First, a Merry Christmas to the Christians and a celebratory solstice to everyone else! Now onto my question, and thanks in advance for any replies:
Let a continuous total charge +q be constrained to move freely on a spherical surface, referring to this from now on as a hollow charged sphere.
If this is statically placed in a constant electric field E, the surface charge will be pushed towards one end so that the total electric field inside returns to zero.
Letting the sphere accelerate, I now have to take into account the self-field being retarded,
.. affecting each charge element in the rest frame of the sphere by a retarded electric field E'. This looks pretty difficult to calculate because I need to know the distribution of the polarized charge around the sphere which in turn will be effectedby the retarded self-field E' acting back on it etc. Which leads me to my question:
What is the charge distribution of a hyperbolically accelerated charged hollow sphere?
Intuitively, I'd expect this retarded self E' to lower the self energy of the polarized static sphere by depolarizing it, thus providing the necessary energy to redistribute the charge: perhaps the charge is evenly redistributed?
Thanks in advance again,
JMcA
On 20/12/23 5:06 PM, john mcandrew wrote:effected by the retarded self-field E' acting back on it etc. Which leads me to my question:
First, a Merry Christmas to the Christians and a celebratory solstice to everyone else! Now onto my question, and thanks in advance for any replies:Merry Christmas John, and to everyone else in the newsgroup!
Let a continuous total charge +q be constrained to move freely on a spherical surface, referring to this from now on as a hollow charged sphere.You have to give more information, especially about the mass (we could
have 3 mass contributions here: the mass of the supporting sphere, that
of the moving charge, and the electromagnetic mass of the surrounding fields).
If this is statically placed in a constant electric field E, the surface charge will be pushed towards one end so that the total electric field inside returns to zero.For that case there is an exact solution (e.g. in Jackson). But that
is strictly speaking not unique. On top of it the charge can be
sloshing back and forward (of course these oscillations will die out eventually, but still, the steady state will then only be reached
at t=infinity)
some time thes
Letting the sphere accelerate, I now have to take into account the self-field being retarded,But do you suddenly release the sphere? Or do you arrange for a more "smooth" onset of the acelleration?
.. affecting each charge element in the rest frame of the sphere by a retarded electric field E'. This looks pretty difficult to calculate because I need to know the distribution of the polarized charge around the sphere which in turn will be
What is the charge distribution of a hyperbolically accelerated charged hollow sphere?Also here the solution will not be unique.. Especially if you release
the sphere quite suddenly. Then there certainly will be a "sloshing
mode" for the charge in the early stages.
Intuitively, I'd expect this retarded self E' to lower the self energy of the polarized static sphere by depolarizing it, thus providing the necessary energy to redistribute the charge: perhaps the charge is evenly redistributed?I would expect the same (after some steady state is reached) but I'm
not sure if this is easy to prove..
Thanks in advance again,
JMcA
--
Jos
On Friday, December 25, 2020 at 11:08:29 AM UTC, Jos Bergervoet wrote:effected by the retarded self-field E' acting back on it etc. Which leads me to my question:
On 20/12/23 5:06 PM, john mcandrew wrote:
Merry Christmas John, and to everyone else in the newsgroup!
First, a Merry Christmas to the Christians and a celebratory solstice to everyone else! Now onto my question, and thanks in advance for any replies:
Let a continuous total charge +q be constrained to move freely on a spherical surface, referring to this from now on as a hollow charged sphere.You have to give more information, especially about the mass (we could
have 3 mass contributions here: the mass of the supporting sphere, that
of the moving charge, and the electromagnetic mass of the surrounding
fields).
If this is statically placed in a constant electric field E, the surface charge will be pushed towards one end so that the total electric field inside returns to zero.For that case there is an exact solution (e.g. in Jackson). But that
is strictly speaking not unique. On top of it the charge can be
sloshing back and forward (of course these oscillations will die out
eventually, but still, the steady state will then only be reached
at t=infinity)
some time thes
Letting the sphere accelerate, I now have to take into account the self-field being retarded,But do you suddenly release the sphere? Or do you arrange for a more
"smooth" onset of the acelleration?
.. affecting each charge element in the rest frame of the sphere by a retarded electric field E'. This looks pretty difficult to calculate because I need to know the distribution of the polarized charge around the sphere which in turn will be
Also here the solution will not be unique.. Especially if you release
What is the charge distribution of a hyperbolically accelerated charged hollow sphere?
the sphere quite suddenly. Then there certainly will be a "sloshing
mode" for the charge in the early stages.
Intuitively, I'd expect this retarded self E' to lower the self energy of the polarized static sphere by depolarizing it, thus providing the necessary energy to redistribute the charge: perhaps the charge is evenly redistributed?I would expect the same (after some steady state is reached) but I'm
not sure if this is easy to prove..
I had in mind the change in the electric field being 'negligible' over the sphere's radius to minimize the sloshing of the charge.
I think the calculation shouldn't be too hard in first assuming the charge distribution in equilibrium is uniform, and then calculating the total electric force on any charge element dq. I'd expect this would be exactly what's required to maintain theconstant charge density or 'rigidness'; so that the trailing edge is given an extra shove compared to the leading edge.
This sloshing of charge seems unavoidable when this charged sphere is moving in a constant magnetic field B; where the electric field in the rest frame of the sphere rotates the charge distribution at a constant angular velocity, if I'm not mistaken.
On 21/01/03 11:13 PM, john mcandrew wrote:effected by the retarded self-field E' acting back on it etc. Which leads me to my question:
On Friday, December 25, 2020 at 11:08:29 AM UTC, Jos Bergervoet wrote:
On 20/12/23 5:06 PM, john mcandrew wrote:
Merry Christmas John, and to everyone else in the newsgroup!
First, a Merry Christmas to the Christians and a celebratory solstice to everyone else! Now onto my question, and thanks in advance for any replies:
Let a continuous total charge +q be constrained to move freely on a spherical surface, referring to this from now on as a hollow charged sphere.You have to give more information, especially about the mass (we could
have 3 mass contributions here: the mass of the supporting sphere, that >> of the moving charge, and the electromagnetic mass of the surrounding
fields).
If this is statically placed in a constant electric field E, the surface charge will be pushed towards one end so that the total electric field inside returns to zero.For that case there is an exact solution (e.g. in Jackson). But that
is strictly speaking not unique. On top of it the charge can be
sloshing back and forward (of course these oscillations will die out
eventually, but still, the steady state will then only be reached
at t=infinity)
some time thes
Letting the sphere accelerate, I now have to take into account the self-field being retarded,But do you suddenly release the sphere? Or do you arrange for a more
"smooth" onset of the acelleration?
.. affecting each charge element in the rest frame of the sphere by a retarded electric field E'. This looks pretty difficult to calculate because I need to know the distribution of the polarized charge around the sphere which in turn will be
the constant charge density or 'rigidness'; so that the trailing edge is given an extra shove compared to the leading edge.Also here the solution will not be unique.. Especially if you release
What is the charge distribution of a hyperbolically accelerated charged hollow sphere?
the sphere quite suddenly. Then there certainly will be a "sloshing
mode" for the charge in the early stages.
Intuitively, I'd expect this retarded self E' to lower the self energy of the polarized static sphere by depolarizing it, thus providing the necessary energy to redistribute the charge: perhaps the charge is evenly redistributed?I would expect the same (after some steady state is reached) but I'm
not sure if this is easy to prove..
I had in mind the change in the electric field being 'negligible' over the sphere's radius to minimize the sloshing of the charge.But you need no pre-existing field at all for the sloshing. It simply
will happen if you bring the charge out of balance.. And of course
it dampens out by radiating (or dissipating if your conductor has resistance).
I think the calculation shouldn't be too hard in first assuming the charge distribution in equilibrium is uniform, and then calculating the total electric force on any charge element dq. I'd expect this would be exactly what's required to maintain
I also think the calculation is simple, once we agree that we can
avoid the sloshing. And that will be an oscilation with a Q-factor
only in the order of 1 (like any dipole antenna). So we can quickly hand-wave it away!
This sloshing of charge seems unavoidable when this charged sphere is moving in a constant magnetic field B; where the electric field in the rest frame of the sphere rotates the charge distribution at a constant angular velocity, if I'm not mistaken."Unavoidable" would only apply if there is incoming radiation of
a certain frequency, for which the sphere will be a receiving antenna.
And I would hesitate to call the sphere's accelerated frame a "rest
frame". But of course it has its own frame which is stationary in
Rindler coordinates.. Acc. to GR's equivalence principle we can view
the sphere as stationary in a frame with a constant gravity field.
And in your case you also have the constant E-field. In fact the two
fields should totally balance each other in the sphere's frame..
This means that there will be a stationary solution with the charge
pulled to one side of the sphere by the E-field and the sphere itself
pulled to the other side by the gravity field. (Assuming the sphere
has non-zero mass). Any additional sloshing will die out soon and just
leave us with this stationary solution.
NB: A pure E-field does not change under Lorentz boost parallel to
the field so it remains what it was, even in the accelerated frame.
On Monday, January 4, 2021 at 12:30:03 PM UTC, Jos Bergervoet wrote:effected by the retarded self-field E' acting back on it etc. Which leads me to my question:
On 21/01/03 11:13 PM, john mcandrew wrote:
On Friday, December 25, 2020 at 11:08:29 AM UTC, Jos Bergervoet wrote:
On 20/12/23 5:06 PM, john mcandrew wrote:
Merry Christmas John, and to everyone else in the newsgroup!
First, a Merry Christmas to the Christians and a celebratory solstice to everyone else! Now onto my question, and thanks in advance for any replies:
Let a continuous total charge +q be constrained to move freely on a spherical surface, referring to this from now on as a hollow charged sphere.You have to give more information, especially about the mass (we could >>>> have 3 mass contributions here: the mass of the supporting sphere, that >>>> of the moving charge, and the electromagnetic mass of the surrounding
fields).
If this is statically placed in a constant electric field E, the surface charge will be pushed towards one end so that the total electric field inside returns to zero.For that case there is an exact solution (e.g. in Jackson). But that
is strictly speaking not unique. On top of it the charge can be
sloshing back and forward (of course these oscillations will die out
eventually, but still, the steady state will then only be reached
at t=infinity)
some time thes
Letting the sphere accelerate, I now have to take into account the self-field being retarded,But do you suddenly release the sphere? Or do you arrange for a more
"smooth" onset of the acelleration?
.. affecting each charge element in the rest frame of the sphere by a retarded electric field E'. This looks pretty difficult to calculate because I need to know the distribution of the polarized charge around the sphere which in turn will be
the constant charge density or 'rigidness'; so that the trailing edge is given an extra shove compared to the leading edge.But you need no pre-existing field at all for the sloshing. It simplyAlso here the solution will not be unique.. Especially if you release
What is the charge distribution of a hyperbolically accelerated charged hollow sphere?
the sphere quite suddenly. Then there certainly will be a "sloshing
mode" for the charge in the early stages.
Intuitively, I'd expect this retarded self E' to lower the self energy of the polarized static sphere by depolarizing it, thus providing the necessary energy to redistribute the charge: perhaps the charge is evenly redistributed?I would expect the same (after some steady state is reached) but I'm
not sure if this is easy to prove..
I had in mind the change in the electric field being 'negligible' over the sphere's radius to minimize the sloshing of the charge.
will happen if you bring the charge out of balance.. And of course
it dampens out by radiating (or dissipating if your conductor has
resistance).
I think the calculation shouldn't be too hard in first assuming the charge distribution in equilibrium is uniform, and then calculating the total electric force on any charge element dq. I'd expect this would be exactly what's required to maintain
I also think the calculation is simple, once we agree that we can
avoid the sloshing. And that will be an oscilation with a Q-factor
only in the order of 1 (like any dipole antenna). So we can quickly
hand-wave it away!
This sloshing of charge seems unavoidable when this charged sphere is moving in a constant magnetic field B; where the electric field in the rest frame of the sphere rotates the charge distribution at a constant angular velocity, if I'm not mistaken."Unavoidable" would only apply if there is incoming radiation of
a certain frequency, for which the sphere will be a receiving antenna.
And I would hesitate to call the sphere's accelerated frame a "rest
frame". But of course it has its own frame which is stationary in
Rindler coordinates.. Acc. to GR's equivalence principle we can view
the sphere as stationary in a frame with a constant gravity field.
And in your case you also have the constant E-field. In fact the two
fields should totally balance each other in the sphere's frame..
This means that there will be a stationary solution with the charge
pulled to one side of the sphere by the E-field and the sphere itself
pulled to the other side by the gravity field. (Assuming the sphere
has non-zero mass). Any additional sloshing will die out soon and just
leave us with this stationary solution.
NB: A pure E-field does not change under Lorentz boost parallel to
the field so it remains what it was, even in the accelerated frame.
I know little about GR,
but I would say that the gravitational field also pulls the charge to the other side, cancelling the polarizing effect of the E field on the charge,
as well as accelerating the sphere depending upon its bare mass: The net effect being a uniformly distributed charge density on the surface of the sphere, with the field lines curved.
Here, I'm interpreting one version of GR's equivalence principle as: an accelerated frame in flat space-time is equivalent to a stationary frame with the equivalent necessary gravitational field.
On 21/01/07 2:14 AM, john mcandrew wrote:effected by the retarded self-field E' acting back on it etc. Which leads me to my question:
On Monday, January 4, 2021 at 12:30:03 PM UTC, Jos Bergervoet wrote:
On 21/01/03 11:13 PM, john mcandrew wrote:
On Friday, December 25, 2020 at 11:08:29 AM UTC, Jos Bergervoet wrote: >>>> On 20/12/23 5:06 PM, john mcandrew wrote:
Merry Christmas John, and to everyone else in the newsgroup!
First, a Merry Christmas to the Christians and a celebratory solstice to everyone else! Now onto my question, and thanks in advance for any replies:
Let a continuous total charge +q be constrained to move freely on a spherical surface, referring to this from now on as a hollow charged sphere.You have to give more information, especially about the mass (we could >>>> have 3 mass contributions here: the mass of the supporting sphere, that >>>> of the moving charge, and the electromagnetic mass of the surrounding >>>> fields).
If this is statically placed in a constant electric field E, the surface charge will be pushed towards one end so that the total electric field inside returns to zero.For that case there is an exact solution (e.g. in Jackson). But that >>>> is strictly speaking not unique. On top of it the charge can be
sloshing back and forward (of course these oscillations will die out >>>> eventually, but still, the steady state will then only be reached
at t=infinity)
some time thes
Letting the sphere accelerate, I now have to take into account the self-field being retarded,But do you suddenly release the sphere? Or do you arrange for a more >>>> "smooth" onset of the acelleration?
.. affecting each charge element in the rest frame of the sphere by a retarded electric field E'. This looks pretty difficult to calculate because I need to know the distribution of the polarized charge around the sphere which in turn will be
the constant charge density or 'rigidness'; so that the trailing edge is given an extra shove compared to the leading edge.But you need no pre-existing field at all for the sloshing. It simplyAlso here the solution will not be unique.. Especially if you release >>>> the sphere quite suddenly. Then there certainly will be a "sloshing >>>> mode" for the charge in the early stages.
What is the charge distribution of a hyperbolically accelerated charged hollow sphere?
Intuitively, I'd expect this retarded self E' to lower the self energy of the polarized static sphere by depolarizing it, thus providing the necessary energy to redistribute the charge: perhaps the charge is evenly redistributed?I would expect the same (after some steady state is reached) but I'm >>>> not sure if this is easy to prove..
I had in mind the change in the electric field being 'negligible' over the sphere's radius to minimize the sloshing of the charge.
will happen if you bring the charge out of balance.. And of course
it dampens out by radiating (or dissipating if your conductor has
resistance).
I think the calculation shouldn't be too hard in first assuming the charge distribution in equilibrium is uniform, and then calculating the total electric force on any charge element dq. I'd expect this would be exactly what's required to maintain
mistaken.I also think the calculation is simple, once we agree that we can
avoid the sloshing. And that will be an oscilation with a Q-factor
only in the order of 1 (like any dipole antenna). So we can quickly
hand-wave it away!
This sloshing of charge seems unavoidable when this charged sphere is moving in a constant magnetic field B; where the electric field in the rest frame of the sphere rotates the charge distribution at a constant angular velocity, if I'm not
"Unavoidable" would only apply if there is incoming radiation of
a certain frequency, for which the sphere will be a receiving antenna.
And I would hesitate to call the sphere's accelerated frame a "rest
frame". But of course it has its own frame which is stationary in
Rindler coordinates.. Acc. to GR's equivalence principle we can view
the sphere as stationary in a frame with a constant gravity field.
And in your case you also have the constant E-field. In fact the two
fields should totally balance each other in the sphere's frame..
This means that there will be a stationary solution with the charge
pulled to one side of the sphere by the E-field and the sphere itself
pulled to the other side by the gravity field. (Assuming the sphere
has non-zero mass). Any additional sloshing will die out soon and just
leave us with this stationary solution.
NB: A pure E-field does not change under Lorentz boost parallel to
the field so it remains what it was, even in the accelerated frame.
I know little about GR,
Amd it isn't needed. Since the equivalence principle in this case
returns everything to a stationary case with a constant gravity
field.
but I would say that the gravitational field also pulls the charge to the other side, cancelling the polarizing effect of the E field on the charge,No, gravity pulls everything to one side, the E-field pulls only
the charge to the other side, so you do get polarization.
as well as accelerating the sphere depending upon its bare mass: The net effect being a uniformly distributed charge density on the surface of the sphere, with the field lines curved.
No, the net effect being a polarized sphere. But of course the
field lines are indeed curved. They would be curved in almost
any solution you construct from either polarized or uniform
charge on the sphere after adding the uniform external E-field.
Here, I'm interpreting one version of GR's equivalence principle as: an accelerated frame in flat space-time is equivalent to a stationary frame with the equivalent necessary gravitational field.Yes, that's the easy part. You then have to reason a bit further
to make sure that the initial uniform E-field remains the same
in this new frame (the same strength and still uniform).
And the solution is a charged sphere hovering. The lift from the
external E-field is balancing gravity. (If it isn't then we haven't
chosen the accelerated frame correctly, it should be co-moving
with the accelerated sphere.)
On Tuesday, January 12, 2021 at 10:25:29 AM UTC, Jos Bergervoet wrote:effected by the retarded self-field E' acting back on it etc. Which leads me to my question:
On 21/01/07 2:14 AM, john mcandrew wrote:
On Monday, January 4, 2021 at 12:30:03 PM UTC, Jos Bergervoet wrote:
On 21/01/03 11:13 PM, john mcandrew wrote:
On Friday, December 25, 2020 at 11:08:29 AM UTC, Jos Bergervoet wrote: >>>>>> On 20/12/23 5:06 PM, john mcandrew wrote:
Merry Christmas John, and to everyone else in the newsgroup!
First, a Merry Christmas to the Christians and a celebratory solstice to everyone else! Now onto my question, and thanks in advance for any replies:
Let a continuous total charge +q be constrained to move freely on a spherical surface, referring to this from now on as a hollow charged sphere.You have to give more information, especially about the mass (we could >>>>>> have 3 mass contributions here: the mass of the supporting sphere, that >>>>>> of the moving charge, and the electromagnetic mass of the surrounding >>>>>> fields).
If this is statically placed in a constant electric field E, the surface charge will be pushed towards one end so that the total electric field inside returns to zero.For that case there is an exact solution (e.g. in Jackson). But that >>>>>> is strictly speaking not unique. On top of it the charge can be
sloshing back and forward (of course these oscillations will die out >>>>>> eventually, but still, the steady state will then only be reached
at t=infinity)
some time thes
Letting the sphere accelerate, I now have to take into account the self-field being retarded,But do you suddenly release the sphere? Or do you arrange for a more >>>>>> "smooth" onset of the acelleration?
.. affecting each charge element in the rest frame of the sphere by a retarded electric field E'. This looks pretty difficult to calculate because I need to know the distribution of the polarized charge around the sphere which in turn will be
the constant charge density or 'rigidness'; so that the trailing edge is given an extra shove compared to the leading edge.But you need no pre-existing field at all for the sloshing. It simplyAlso here the solution will not be unique.. Especially if you release >>>>>> the sphere quite suddenly. Then there certainly will be a "sloshing >>>>>> mode" for the charge in the early stages.
What is the charge distribution of a hyperbolically accelerated charged hollow sphere?
Intuitively, I'd expect this retarded self E' to lower the self energy of the polarized static sphere by depolarizing it, thus providing the necessary energy to redistribute the charge: perhaps the charge is evenly redistributed?I would expect the same (after some steady state is reached) but I'm >>>>>> not sure if this is easy to prove..
I had in mind the change in the electric field being 'negligible' over the sphere's radius to minimize the sloshing of the charge.
will happen if you bring the charge out of balance.. And of course
it dampens out by radiating (or dissipating if your conductor has
resistance).
I think the calculation shouldn't be too hard in first assuming the charge distribution in equilibrium is uniform, and then calculating the total electric force on any charge element dq. I'd expect this would be exactly what's required to maintain
mistaken.I also think the calculation is simple, once we agree that we can
avoid the sloshing. And that will be an oscilation with a Q-factor
only in the order of 1 (like any dipole antenna). So we can quickly
hand-wave it away!
This sloshing of charge seems unavoidable when this charged sphere is moving in a constant magnetic field B; where the electric field in the rest frame of the sphere rotates the charge distribution at a constant angular velocity, if I'm not
"Unavoidable" would only apply if there is incoming radiation of
a certain frequency, for which the sphere will be a receiving antenna.
And I would hesitate to call the sphere's accelerated frame a "rest
frame". But of course it has its own frame which is stationary in
Rindler coordinates.. Acc. to GR's equivalence principle we can view
the sphere as stationary in a frame with a constant gravity field.
And in your case you also have the constant E-field. In fact the two
fields should totally balance each other in the sphere's frame..
This means that there will be a stationary solution with the charge
pulled to one side of the sphere by the E-field and the sphere itself
pulled to the other side by the gravity field. (Assuming the sphere
has non-zero mass). Any additional sloshing will die out soon and just >>>> leave us with this stationary solution.
NB: A pure E-field does not change under Lorentz boost parallel to
the field so it remains what it was, even in the accelerated frame.
I know little about GR,
Amd it isn't needed. Since the equivalence principle in this case
returns everything to a stationary case with a constant gravity
field.
I'm assuming you mean GR rather than the E field isn't needed, and Newtonian gravity is adequate?
have a varying effect upon the surface charge, causing it to at least polarize to a degree.but I would say that the gravitational field also pulls the charge to the other side, cancelling the polarizing effect of the E field on the charge,No, gravity pulls everything to one side, the E-field pulls only
the charge to the other side, so you do get polarization.
as well as accelerating the sphere depending upon its bare mass: The net effect being a uniformly distributed charge density on the surface of the sphere, with the field lines curved.
No, the net effect being a polarized sphere. But of course the
field lines are indeed curved. They would be curved in almost
any solution you construct from either polarized or uniform
charge on the sphere after adding the uniform external E-field.
This is what confuses me: the polarizing of the surface charge by the E field means the corresponding energy density and therefore equivalent mass at each point also varies upon the spherical surface. Hence I'd expect the gravitational field to now
Yes, that's the easy part. You then have to reason a bit further
Here, I'm interpreting one version of GR's equivalence principle as: an accelerated frame in flat space-time is equivalent to a stationary frame with the equivalent necessary gravitational field.
to make sure that the initial uniform E-field remains the same
in this new frame (the same strength and still uniform).
And the solution is a charged sphere hovering. The lift from the
external E-field is balancing gravity. (If it isn't then we haven't
chosen the accelerated frame correctly, it should be co-moving
with the accelerated sphere.)
I agree that the charged sphere is hovering, but also that the charge remains uniformly distributed
as in the equivalent accelerated frame without the gravitational field, which agrees with the equivalence principle here if I'm not mistaken.
On 20/12/23 5:06 PM, john mcandrew wrote:effected by the retarded self-field E' acting back on it etc. Which leads me to my question:
First, a Merry Christmas to the Christians and a celebratory solstice to everyone else! Now onto my question, and thanks in advance for any replies:Merry Christmas John, and to everyone else in the newsgroup!
Let a continuous total charge +q be constrained to move freely on a spherical surface, referring to this from now on as a hollow charged sphere.You have to give more information, especially about the mass (we could
have 3 mass contributions here: the mass of the supporting sphere, that
of the moving charge, and the electromagnetic mass of the surrounding fields).
If this is statically placed in a constant electric field E, the surface charge will be pushed towards one end so that the total electric field inside returns to zero.For that case there is an exact solution (e.g. in Jackson). But that
is strictly speaking not unique. On top of it the charge can be
sloshing back and forward (of course these oscillations will die out eventually, but still, the steady state will then only be reached
at t=infinity)
some time thes
Letting the sphere accelerate, I now have to take into account the self-field being retarded,But do you suddenly release the sphere? Or do you arrange for a more "smooth" onset of the acelleration?
.. affecting each charge element in the rest frame of the sphere by a retarded electric field E'. This looks pretty difficult to calculate because I need to know the distribution of the polarized charge around the sphere which in turn will be
What is the charge distribution of a hyperbolically accelerated charged hollow sphere?Also here the solution will not be unique.. Especially if you release
the sphere quite suddenly. Then there certainly will be a "sloshing
mode" for the charge in the early stages.
Intuitively, I'd expect this retarded self E' to lower the self energy of the polarized static sphere by depolarizing it, thus providing the necessary energy to redistribute the charge: perhaps the charge is evenly redistributed?I would expect the same (after some steady state is reached) but I'm
not sure if this is easy to prove..
On Friday, December 25, 2020 at 11:08:29 AM UTC, Jos Bergervoet wrote:effected by the retarded self-field E' acting back on it etc. Which leads me to my question:
On 20/12/23 5:06 PM, john mcandrew wrote:
Merry Christmas John, and to everyone else in the newsgroup!
First, a Merry Christmas to the Christians and a celebratory solstice to everyone else! Now onto my question, and thanks in advance for any replies:
Let a continuous total charge +q be constrained to move freely on a spherical surface, referring to this from now on as a hollow charged sphere.You have to give more information, especially about the mass (we could
have 3 mass contributions here: the mass of the supporting sphere, that
of the moving charge, and the electromagnetic mass of the surrounding
fields).
If this is statically placed in a constant electric field E, the surface charge will be pushed towards one end so that the total electric field inside returns to zero.For that case there is an exact solution (e.g. in Jackson). But that
is strictly speaking not unique. On top of it the charge can be
sloshing back and forward (of course these oscillations will die out
eventually, but still, the steady state will then only be reached
at t=infinity)
some time thes
Letting the sphere accelerate, I now have to take into account the self-field being retarded,But do you suddenly release the sphere? Or do you arrange for a more
"smooth" onset of the acelleration?
.. affecting each charge element in the rest frame of the sphere by a retarded electric field E'. This looks pretty difficult to calculate because I need to know the distribution of the polarized charge around the sphere which in turn will be
Also here the solution will not be unique.. Especially if you release
What is the charge distribution of a hyperbolically accelerated charged hollow sphere?
the sphere quite suddenly. Then there certainly will be a "sloshing
mode" for the charge in the early stages.
Intuitively, I'd expect this retarded self E' to lower the self energy of the polarized static sphere by depolarizing it, thus providing the necessary energy to redistribute the charge: perhaps the charge is evenly redistributed?I would expect the same (after some steady state is reached) but I'm
not sure if this is easy to prove..
You're agreeing with me here,
in that the charge will end up evenly distributed in the steady state when the sphere is hyperbolically accelerated: the applied E field polarizing the charge and the retarded E' field depolarizing it, cancelling one another exactly.
But doesn't this imply the charge is also uniformly distributed in the equivalent case with a gravitational field replacing the accelerated frame?
On 21/01/14 12:26 AM, john mcandrew wrote:effected by the retarded self-field E' acting back on it etc. Which leads me to my question:
On Friday, December 25, 2020 at 11:08:29 AM UTC, Jos Bergervoet wrote:
On 20/12/23 5:06 PM, john mcandrew wrote:
Merry Christmas John, and to everyone else in the newsgroup!
First, a Merry Christmas to the Christians and a celebratory solstice to everyone else! Now onto my question, and thanks in advance for any replies:
Let a continuous total charge +q be constrained to move freely on a spherical surface, referring to this from now on as a hollow charged sphere.You have to give more information, especially about the mass (we could
have 3 mass contributions here: the mass of the supporting sphere, that >> of the moving charge, and the electromagnetic mass of the surrounding
fields).
If this is statically placed in a constant electric field E, the surface charge will be pushed towards one end so that the total electric field inside returns to zero.For that case there is an exact solution (e.g. in Jackson). But that
is strictly speaking not unique. On top of it the charge can be
sloshing back and forward (of course these oscillations will die out
eventually, but still, the steady state will then only be reached
at t=infinity)
some time thes
Letting the sphere accelerate, I now have to take into account the self-field being retarded,But do you suddenly release the sphere? Or do you arrange for a more
"smooth" onset of the acelleration?
.. affecting each charge element in the rest frame of the sphere by a retarded electric field E'. This looks pretty difficult to calculate because I need to know the distribution of the polarized charge around the sphere which in turn will be
Also here the solution will not be unique.. Especially if you release
What is the charge distribution of a hyperbolically accelerated charged hollow sphere?
the sphere quite suddenly. Then there certainly will be a "sloshing
mode" for the charge in the early stages.
Intuitively, I'd expect this retarded self E' to lower the self energy of the polarized static sphere by depolarizing it, thus providing the necessary energy to redistribute the charge: perhaps the charge is evenly redistributed?I would expect the same (after some steady state is reached) but I'm
not sure if this is easy to prove..
You're agreeing with me here,Agreeing with you that the lowest energy will eventually be
reached, also agreeing that depolarization will happen after
initial polarization, but it will not be depolarized back
to zero. After the E-field is (magically) switched on and the
sphere starts to accelerate, I would (intuitively) expect the
polarization vs. time to be something like this:
* *
* * * * * *
* * *
* * *
0*
Of course the 'sloshing' will consist of infinitely many
oscillations, decreasing exponentially in strength.
in that the charge will end up evenly distributed in the steady state when the sphere is hyperbolically accelerated: the applied E field polarizing the charge and the retarded E' field depolarizing it, cancelling one another exactly.It isn't what I would intuitively expect, I'd expect that as
long as the acceleration lasts, the charge is being pulled a
little bit ahead of the sphere's centre, so it remains
polarized all the time.
But doesn't this imply the charge is also uniformly distributed in the equivalent case with a gravitational field replacing the accelerated frame?My statement would imply that the sphere is *also polarized*
when hovering in the gravitational field. Also there, the
charge is pulled upwards by the E-field and will be off-center.
And in the latter case I am much more confident that this is
easy to prove! (Of course we then do have to prove in addition
that the equivalence principle actually applies here, but still
this reduces the original problem to proving two sub-problems.)
--
Jos
On Thursday, January 14, 2021 at 9:20:16 PM UTC, Jos Bergervoet wrote:effected by the retarded self-field E' acting back on it etc. Which leads me to my question:
On 21/01/14 12:26 AM, john mcandrew wrote:
On Friday, December 25, 2020 at 11:08:29 AM UTC, Jos Bergervoet wrote:
On 20/12/23 5:06 PM, john mcandrew wrote:
Merry Christmas John, and to everyone else in the newsgroup!
First, a Merry Christmas to the Christians and a celebratory solstice to everyone else! Now onto my question, and thanks in advance for any replies:
Let a continuous total charge +q be constrained to move freely on a spherical surface, referring to this from now on as a hollow charged sphere.You have to give more information, especially about the mass (we could >>>> have 3 mass contributions here: the mass of the supporting sphere, that >>>> of the moving charge, and the electromagnetic mass of the surrounding
fields).
If this is statically placed in a constant electric field E, the surface charge will be pushed towards one end so that the total electric field inside returns to zero.For that case there is an exact solution (e.g. in Jackson). But that
is strictly speaking not unique. On top of it the charge can be
sloshing back and forward (of course these oscillations will die out
eventually, but still, the steady state will then only be reached
at t=infinity)
some time thes
Letting the sphere accelerate, I now have to take into account the self-field being retarded,But do you suddenly release the sphere? Or do you arrange for a more
"smooth" onset of the acelleration?
.. affecting each charge element in the rest frame of the sphere by a retarded electric field E'. This looks pretty difficult to calculate because I need to know the distribution of the polarized charge around the sphere which in turn will be
Agreeing with you that the lowest energy will eventually beAlso here the solution will not be unique.. Especially if you release
What is the charge distribution of a hyperbolically accelerated charged hollow sphere?
the sphere quite suddenly. Then there certainly will be a "sloshing
mode" for the charge in the early stages.
Intuitively, I'd expect this retarded self E' to lower the self energy of the polarized static sphere by depolarizing it, thus providing the necessary energy to redistribute the charge: perhaps the charge is evenly redistributed?I would expect the same (after some steady state is reached) but I'm
not sure if this is easy to prove..
You're agreeing with me here,
reached, also agreeing that depolarization will happen after
initial polarization, but it will not be depolarized back
to zero. After the E-field is (magically) switched on and the
sphere starts to accelerate, I would (intuitively) expect the
polarization vs. time to be something like this:
* *
* * * * * *
* * *
* * *
0*
Of course the 'sloshing' will consist of infinitely many
oscillations, decreasing exponentially in strength.
in that the charge will end up evenly distributed in the steady state when the sphere is hyperbolically accelerated: the applied E field polarizing the charge and the retarded E' field depolarizing it, cancelling one another exactly.It isn't what I would intuitively expect, I'd expect that as
long as the acceleration lasts, the charge is being pulled a
little bit ahead of the sphere's centre, so it remains
polarized all the time.
But doesn't this imply the charge is also uniformly distributed in the equivalent case with a gravitational field replacing the accelerated frame?My statement would imply that the sphere is *also polarized*
when hovering in the gravitational field. Also there, the
charge is pulled upwards by the E-field and will be off-center.
And in the latter case I am much more confident that this is
easy to prove! (Of course we then do have to prove in addition
that the equivalence principle actually applies here, but still
this reduces the original problem to proving two sub-problems.)
--
Jos
I agree with the above for a sphere with a finite bare mass that dominates the electromagnetic mass. And I also agree with what you've written elsewhere in this thread:
https://groups.google.com/g/sci.physics.electromag/c/JTPBOZ3W4wU/m/bqoSBw5fCQAJ
"The equivalence principle will not change the charge. It may deform
the sphere by Lorentz contraction when going from one frame to the
other, but in thic case that won't change a non-uniform distribution
into a uniform one.
And the charge will be non-uniform. Except in the case when the sphere
has zero mass (and all mass is in the charged particles and in the
E-field density, and no mass in the rigid sphere. But that is
unphysical)."
Would I be correct in saying that you believe the polarizing and depolarizing effects on the surface charge approach cancelling one another as the electromagnetic mass >> bare mass?
On 21/01/15 1:25 AM, john mcandrew wrote:effected by the retarded self-field E' acting back on it etc. Which leads me to my question:
On Thursday, January 14, 2021 at 9:20:16 PM UTC, Jos Bergervoet wrote:
On 21/01/14 12:26 AM, john mcandrew wrote:
On Friday, December 25, 2020 at 11:08:29 AM UTC, Jos Bergervoet wrote: >>>> On 20/12/23 5:06 PM, john mcandrew wrote:
Merry Christmas John, and to everyone else in the newsgroup!
First, a Merry Christmas to the Christians and a celebratory solstice to everyone else! Now onto my question, and thanks in advance for any replies:
Let a continuous total charge +q be constrained to move freely on a spherical surface, referring to this from now on as a hollow charged sphere.You have to give more information, especially about the mass (we could >>>> have 3 mass contributions here: the mass of the supporting sphere, that >>>> of the moving charge, and the electromagnetic mass of the surrounding >>>> fields).
If this is statically placed in a constant electric field E, the surface charge will be pushed towards one end so that the total electric field inside returns to zero.For that case there is an exact solution (e.g. in Jackson). But that >>>> is strictly speaking not unique. On top of it the charge can be
sloshing back and forward (of course these oscillations will die out >>>> eventually, but still, the steady state will then only be reached
at t=infinity)
some time thes
Letting the sphere accelerate, I now have to take into account the self-field being retarded,But do you suddenly release the sphere? Or do you arrange for a more >>>> "smooth" onset of the acelleration?
.. affecting each charge element in the rest frame of the sphere by a retarded electric field E'. This looks pretty difficult to calculate because I need to know the distribution of the polarized charge around the sphere which in turn will be
Agreeing with you that the lowest energy will eventually beAlso here the solution will not be unique.. Especially if you release >>>> the sphere quite suddenly. Then there certainly will be a "sloshing >>>> mode" for the charge in the early stages.
What is the charge distribution of a hyperbolically accelerated charged hollow sphere?
Intuitively, I'd expect this retarded self E' to lower the self energy of the polarized static sphere by depolarizing it, thus providing the necessary energy to redistribute the charge: perhaps the charge is evenly redistributed?I would expect the same (after some steady state is reached) but I'm >>>> not sure if this is easy to prove..
You're agreeing with me here,
reached, also agreeing that depolarization will happen after
initial polarization, but it will not be depolarized back
to zero. After the E-field is (magically) switched on and the
sphere starts to accelerate, I would (intuitively) expect the
polarization vs. time to be something like this:
* *
* * * * * *
* * *
* * *
0*
Of course the 'sloshing' will consist of infinitely many
oscillations, decreasing exponentially in strength.
in that the charge will end up evenly distributed in the steady state when the sphere is hyperbolically accelerated: the applied E field polarizing the charge and the retarded E' field depolarizing it, cancelling one another exactly.It isn't what I would intuitively expect, I'd expect that as
long as the acceleration lasts, the charge is being pulled a
little bit ahead of the sphere's centre, so it remains
polarized all the time.
But doesn't this imply the charge is also uniformly distributed in the equivalent case with a gravitational field replacing the accelerated frame?My statement would imply that the sphere is *also polarized*
when hovering in the gravitational field. Also there, the
charge is pulled upwards by the E-field and will be off-center.
And in the latter case I am much more confident that this is
easy to prove! (Of course we then do have to prove in addition
that the equivalence principle actually applies here, but still
this reduces the original problem to proving two sub-problems.)
--
Jos
I agree with the above for a sphere with a finite bare mass that dominates the electromagnetic mass. And I also agree with what you've written elsewhere in this thread:
https://groups.google.com/g/sci.physics.electromag/c/JTPBOZ3W4wU/m/bqoSBw5fCQAJ
"The equivalence principle will not change the charge. It may deform
the sphere by Lorentz contraction when going from one frame to the
other, but in thic case that won't change a non-uniform distribution
into a uniform one.
And the charge will be non-uniform. Except in the case when the sphere
has zero mass (and all mass is in the charged particles and in the
E-field density, and no mass in the rigid sphere. But that is unphysical)."
Would I be correct in saying that you believe the polarizing and depolarizing effects on the surface charge approach cancelling one another as the electromagnetic mass >> bare mass?When we discussed what could "intuitively" be expected, I only
considered decent, simple, 19th-century electrostatics, with
physical objects that exist in everyday life!
With a mass-zero sphere (and mass-less charge carriers as well)
things might start to escape our range of intuition.. I would
not be sure whether the pull of gravity will actually change the
direction of field lines, for instance, if the system is hovering
in a constant gravity field. The problem is that, even if the
equivalence principle allows us to use that frame, I do not know
how Maxwell's equations might change in a non-inertial frame. (It
probably can be looked up on Wikipedia, but intuitively I would
just expect the unexpected.)
For safety, I would go back to the original frame, where we know
Maxwell, and try to solve it there. But the combination of E-
and B-fields and moving charges make this complicated of course.
The precise shape of the field is not obvious to begin with, and
since you ask whether some things might be precisely "cancelled",
you would really need precise information!
Alternatively, we can work in the sphere's own stationary frame
and *assume* that Maxwell is unaltered for electrostatics in a
gravity field. In that case your proposed solution with uniform
charge does not seem to be stable. There is an upward pull of the
E-field on the charge carriers (which have no mass so they are
not pulled down by gravity) and there must be a downward pull of
gravity on the E-field energy density around the sphere (which
contains all the mass now). So intuitively we might still expect
the charge to move off centre (upwards) and the E-field lines to
be drooping downwards.. but the latter is incompatible with the
starting assumption that electrostatics is unaltered by gravity.
(So probably it isn't!)
On Sunday, January 17, 2021 at 10:50:02 AM UTC, Jos Bergervoet wrote:effected by the retarded self-field E' acting back on it etc. Which leads me to my question:
On 21/01/15 1:25 AM, john mcandrew wrote:
On Thursday, January 14, 2021 at 9:20:16 PM UTC, Jos Bergervoet wrote:
On 21/01/14 12:26 AM, john mcandrew wrote:
On Friday, December 25, 2020 at 11:08:29 AM UTC, Jos Bergervoet wrote: >>>>>> On 20/12/23 5:06 PM, john mcandrew wrote:
Merry Christmas John, and to everyone else in the newsgroup!
First, a Merry Christmas to the Christians and a celebratory solstice to everyone else! Now onto my question, and thanks in advance for any replies:
Let a continuous total charge +q be constrained to move freely on a spherical surface, referring to this from now on as a hollow charged sphere.You have to give more information, especially about the mass (we could >>>>>> have 3 mass contributions here: the mass of the supporting sphere, that >>>>>> of the moving charge, and the electromagnetic mass of the surrounding >>>>>> fields).
If this is statically placed in a constant electric field E, the surface charge will be pushed towards one end so that the total electric field inside returns to zero.For that case there is an exact solution (e.g. in Jackson). But that >>>>>> is strictly speaking not unique. On top of it the charge can be
sloshing back and forward (of course these oscillations will die out >>>>>> eventually, but still, the steady state will then only be reached
at t=infinity)
some time thes
Letting the sphere accelerate, I now have to take into account the self-field being retarded,But do you suddenly release the sphere? Or do you arrange for a more >>>>>> "smooth" onset of the acelleration?
.. affecting each charge element in the rest frame of the sphere by a retarded electric field E'. This looks pretty difficult to calculate because I need to know the distribution of the polarized charge around the sphere which in turn will be
field. As the bare mass --> 0, the acceleration increases causing the retarded field to have an increasing effect on the charge distribution, noting that I only have to worry about the retarded electric field for steady state conditions in the sphere'sWhen we discussed what could "intuitively" be expected, I onlyAgreeing with you that the lowest energy will eventually beAlso here the solution will not be unique.. Especially if you release >>>>>> the sphere quite suddenly. Then there certainly will be a "sloshing >>>>>> mode" for the charge in the early stages.
What is the charge distribution of a hyperbolically accelerated charged hollow sphere?
Intuitively, I'd expect this retarded self E' to lower the self energy of the polarized static sphere by depolarizing it, thus providing the necessary energy to redistribute the charge: perhaps the charge is evenly redistributed?I would expect the same (after some steady state is reached) but I'm >>>>>> not sure if this is easy to prove..
You're agreeing with me here,
reached, also agreeing that depolarization will happen after
initial polarization, but it will not be depolarized back
to zero. After the E-field is (magically) switched on and the
sphere starts to accelerate, I would (intuitively) expect the
polarization vs. time to be something like this:
* *
* * * * * *
* * *
* * *
0*
Of course the 'sloshing' will consist of infinitely many
oscillations, decreasing exponentially in strength.
in that the charge will end up evenly distributed in the steady state when the sphere is hyperbolically accelerated: the applied E field polarizing the charge and the retarded E' field depolarizing it, cancelling one another exactly.It isn't what I would intuitively expect, I'd expect that as
long as the acceleration lasts, the charge is being pulled a
little bit ahead of the sphere's centre, so it remains
polarized all the time.
But doesn't this imply the charge is also uniformly distributed in the equivalent case with a gravitational field replacing the accelerated frame?My statement would imply that the sphere is *also polarized*
when hovering in the gravitational field. Also there, the
charge is pulled upwards by the E-field and will be off-center.
And in the latter case I am much more confident that this is
easy to prove! (Of course we then do have to prove in addition
that the equivalence principle actually applies here, but still
this reduces the original problem to proving two sub-problems.)
--
Jos
I agree with the above for a sphere with a finite bare mass that dominates the electromagnetic mass. And I also agree with what you've written elsewhere in this thread:
https://groups.google.com/g/sci.physics.electromag/c/JTPBOZ3W4wU/m/bqoSBw5fCQAJ
"The equivalence principle will not change the charge. It may deform
the sphere by Lorentz contraction when going from one frame to the
other, but in thic case that won't change a non-uniform distribution
into a uniform one.
And the charge will be non-uniform. Except in the case when the sphere
has zero mass (and all mass is in the charged particles and in the
E-field density, and no mass in the rigid sphere. But that is
unphysical)."
Would I be correct in saying that you believe the polarizing and depolarizing effects on the surface charge approach cancelling one another as the electromagnetic mass >> bare mass?
considered decent, simple, 19th-century electrostatics, with
physical objects that exist in everyday life!
With a mass-zero sphere (and mass-less charge carriers as well)
things might start to escape our range of intuition.. I would
not be sure whether the pull of gravity will actually change the
direction of field lines, for instance, if the system is hovering
in a constant gravity field. The problem is that, even if the
equivalence principle allows us to use that frame, I do not know
how Maxwell's equations might change in a non-inertial frame. (It
probably can be looked up on Wikipedia, but intuitively I would
just expect the unexpected.)
For safety, I would go back to the original frame, where we know
Maxwell, and try to solve it there. But the combination of E-
and B-fields and moving charges make this complicated of course.
The precise shape of the field is not obvious to begin with, and
since you ask whether some things might be precisely "cancelled",
you would really need precise information!
Initially, I can imagine the sphere to have a large bare mass compared to the electromagnetic so that it's 'gently" accelerated. The surface will obviously be an equipotential with a non-uniform charge distribution giving rise to an internal 0 electric
For hyperbolic motion, the LAD equation for a point charge has the Shott and radiation terms cancelling one another; the Shott term being the "acceleration energy" above. So likewise I'm confident of this cancelling for a charge sphere also, althoughto be sure I need to calculate it of course.
Alternatively, we can work in the sphere's own stationary frame
and *assume* that Maxwell is unaltered for electrostatics in a
gravity field. In that case your proposed solution with uniform
charge does not seem to be stable. There is an upward pull of the
E-field on the charge carriers (which have no mass so they are
not pulled down by gravity) and there must be a downward pull of
gravity on the E-field energy density around the sphere (which
contains all the mass now). So intuitively we might still expect
the charge to move off centre (upwards) and the E-field lines to
be drooping downwards.. but the latter is incompatible with the
starting assumption that electrostatics is unaltered by gravity.
(So probably it isn't!)
GR looks like a minefield to me even for the experts which is why I try to keep away from it.
...
Resolution by Rohrlich https://en.wikipedia.org/wiki/Paradox_of_radiation_of_charged_particles_in_a_gravitational_field#Resolution_by_Rohrlich
On 21/01/21 12:35 AM, john mcandrew wrote:effected by the retarded self-field E' acting back on it etc. Which leads me to my question:
On Sunday, January 17, 2021 at 10:50:02 AM UTC, Jos Bergervoet wrote:
On 21/01/15 1:25 AM, john mcandrew wrote:
On Thursday, January 14, 2021 at 9:20:16 PM UTC, Jos Bergervoet wrote: >>>> On 21/01/14 12:26 AM, john mcandrew wrote:
On Friday, December 25, 2020 at 11:08:29 AM UTC, Jos Bergervoet wrote: >>>>>> On 20/12/23 5:06 PM, john mcandrew wrote:
Merry Christmas John, and to everyone else in the newsgroup!
First, a Merry Christmas to the Christians and a celebratory solstice to everyone else! Now onto my question, and thanks in advance for any replies:
Let a continuous total charge +q be constrained to move freely on a spherical surface, referring to this from now on as a hollow charged sphere.You have to give more information, especially about the mass (we could
have 3 mass contributions here: the mass of the supporting sphere, that
of the moving charge, and the electromagnetic mass of the surrounding >>>>>> fields).
If this is statically placed in a constant electric field E, the surface charge will be pushed towards one end so that the total electric field inside returns to zero.For that case there is an exact solution (e.g. in Jackson). But that >>>>>> is strictly speaking not unique. On top of it the charge can be >>>>>> sloshing back and forward (of course these oscillations will die out >>>>>> eventually, but still, the steady state will then only be reached >>>>>> at t=infinity)
some time thes
Letting the sphere accelerate, I now have to take into account the self-field being retarded,But do you suddenly release the sphere? Or do you arrange for a more >>>>>> "smooth" onset of the acelleration?
.. affecting each charge element in the rest frame of the sphere by a retarded electric field E'. This looks pretty difficult to calculate because I need to know the distribution of the polarized charge around the sphere which in turn will be
electric field. As the bare mass --> 0, the acceleration increases causing the retarded field to have an increasing effect on the charge distribution, noting that I only have to worry about the retarded electric field for steady state conditions in theWhen we discussed what could "intuitively" be expected, I onlyAgreeing with you that the lowest energy will eventually beAlso here the solution will not be unique.. Especially if you release >>>>>> the sphere quite suddenly. Then there certainly will be a "sloshing >>>>>> mode" for the charge in the early stages.
What is the charge distribution of a hyperbolically accelerated charged hollow sphere?
Intuitively, I'd expect this retarded self E' to lower the self energy of the polarized static sphere by depolarizing it, thus providing the necessary energy to redistribute the charge: perhaps the charge is evenly redistributed?I would expect the same (after some steady state is reached) but I'm >>>>>> not sure if this is easy to prove..
You're agreeing with me here,
reached, also agreeing that depolarization will happen after
initial polarization, but it will not be depolarized back
to zero. After the E-field is (magically) switched on and the
sphere starts to accelerate, I would (intuitively) expect the
polarization vs. time to be something like this:
* *
* * * * * *
* * *
* * *
0*
Of course the 'sloshing' will consist of infinitely many
oscillations, decreasing exponentially in strength.
in that the charge will end up evenly distributed in the steady state when the sphere is hyperbolically accelerated: the applied E field polarizing the charge and the retarded E' field depolarizing it, cancelling one another exactly.It isn't what I would intuitively expect, I'd expect that as
long as the acceleration lasts, the charge is being pulled a
little bit ahead of the sphere's centre, so it remains
polarized all the time.
But doesn't this imply the charge is also uniformly distributed in the equivalent case with a gravitational field replacing the accelerated frame?My statement would imply that the sphere is *also polarized*
when hovering in the gravitational field. Also there, the
charge is pulled upwards by the E-field and will be off-center.
And in the latter case I am much more confident that this is
easy to prove! (Of course we then do have to prove in addition
that the equivalence principle actually applies here, but still
this reduces the original problem to proving two sub-problems.)
--
Jos
I agree with the above for a sphere with a finite bare mass that dominates the electromagnetic mass. And I also agree with what you've written elsewhere in this thread:
https://groups.google.com/g/sci.physics.electromag/c/JTPBOZ3W4wU/m/bqoSBw5fCQAJ
"The equivalence principle will not change the charge. It may deform
the sphere by Lorentz contraction when going from one frame to the
other, but in thic case that won't change a non-uniform distribution
into a uniform one.
And the charge will be non-uniform. Except in the case when the sphere >>> has zero mass (and all mass is in the charged particles and in the
E-field density, and no mass in the rigid sphere. But that is
unphysical)."
Would I be correct in saying that you believe the polarizing and depolarizing effects on the surface charge approach cancelling one another as the electromagnetic mass >> bare mass?
considered decent, simple, 19th-century electrostatics, with
physical objects that exist in everyday life!
With a mass-zero sphere (and mass-less charge carriers as well)
things might start to escape our range of intuition.. I would
not be sure whether the pull of gravity will actually change the
direction of field lines, for instance, if the system is hovering
in a constant gravity field. The problem is that, even if the
equivalence principle allows us to use that frame, I do not know
how Maxwell's equations might change in a non-inertial frame. (It
probably can be looked up on Wikipedia, but intuitively I would
just expect the unexpected.)
For safety, I would go back to the original frame, where we know
Maxwell, and try to solve it there. But the combination of E-
and B-fields and moving charges make this complicated of course.
The precise shape of the field is not obvious to begin with, and
since you ask whether some things might be precisely "cancelled",
you would really need precise information!
Initially, I can imagine the sphere to have a large bare mass compared to the electromagnetic so that it's 'gently" accelerated. The surface will obviously be an equipotential with a non-uniform charge distribution giving rise to an internal 0
to be sure I need to calculate it of course.For hyperbolic motion, the LAD equation for a point charge has the Shott and radiation terms cancelling one another; the Shott term being the "acceleration energy" above. So likewise I'm confident of this cancelling for a charge sphere also, although
Alternatively, we can work in the sphere's own stationary frame
and *assume* that Maxwell is unaltered for electrostatics in a
gravity field. In that case your proposed solution with uniform
charge does not seem to be stable. There is an upward pull of the
E-field on the charge carriers (which have no mass so they are
not pulled down by gravity) and there must be a downward pull of
gravity on the E-field energy density around the sphere (which
contains all the mass now). So intuitively we might still expect
the charge to move off centre (upwards) and the E-field lines to
be drooping downwards.. but the latter is incompatible with the
starting assumption that electrostatics is unaltered by gravity.
(So probably it isn't!)
GR looks like a minefield to me even for the experts which is why I try to keep away from it.But that makes it intriguing to see which mines there are in any
given case! In the case here it is the EM-field's behavior in a non- inertial frame, i.e. with constant gravity present. In that case
Maxwell's equations, and even simple Coulomb electrostatics, do not
apply unaltered!
As you can read in Wikipedia's Rohrlich paragraph below, "Maxwell
equations do not hold" and a bit further "the gravitational field
slightly distorts the Coulomb field".
Of course we also know that combined E- and B-fields, like a moving
wave packet, would not propagate according to plain Maxwell, because
that would be rectilinear motion of the packet, and we know that
light is bent downwards by gravity. But it seems even electrostatics
is affected..
...All this does not answer your question whether the charge will be
Resolution by Rohrlich https://en.wikipedia.org/wiki/Paradox_of_radiation_of_charged_particles_in_a_gravitational_field#Resolution_by_Rohrlich
uniformly distributed in the sphere's (non-inertial) frame, but the
fact that gravity "slightly distorts" the ordinary electrostatic
situation, as stated, would make me believe (intuitively) that the
charge is not uniform any more..
--
Jos
On Saturday, January 23, 2021 at 10:30:02 PM UTC, Jos Bergervoet wrote:effected by the retarded self-field E' acting back on it etc. Which leads me to my question:
On 21/01/21 12:35 AM, john mcandrew wrote:
On Sunday, January 17, 2021 at 10:50:02 AM UTC, Jos Bergervoet wrote:
On 21/01/15 1:25 AM, john mcandrew wrote:
On Thursday, January 14, 2021 at 9:20:16 PM UTC, Jos Bergervoet wrote: >>>> On 21/01/14 12:26 AM, john mcandrew wrote:
On Friday, December 25, 2020 at 11:08:29 AM UTC, Jos Bergervoet wrote:
On 20/12/23 5:06 PM, john mcandrew wrote:
Merry Christmas John, and to everyone else in the newsgroup! >>>>>>> Let a continuous total charge +q be constrained to move freely on a spherical surface, referring to this from now on as a hollow charged sphere.
First, a Merry Christmas to the Christians and a celebratory solstice to everyone else! Now onto my question, and thanks in advance for any replies:
You have to give more information, especially about the mass (we could
have 3 mass contributions here: the mass of the supporting sphere, that
of the moving charge, and the electromagnetic mass of the surrounding
fields).
If this is statically placed in a constant electric field E, the surface charge will be pushed towards one end so that the total electric field inside returns to zero.For that case there is an exact solution (e.g. in Jackson). But that
is strictly speaking not unique. On top of it the charge can be >>>>>> sloshing back and forward (of course these oscillations will die out
eventually, but still, the steady state will then only be reached >>>>>> at t=infinity)
some time thes
Letting the sphere accelerate, I now have to take into account the self-field being retarded,But do you suddenly release the sphere? Or do you arrange for a more
"smooth" onset of the acelleration?
.. affecting each charge element in the rest frame of the sphere by a retarded electric field E'. This looks pretty difficult to calculate because I need to know the distribution of the polarized charge around the sphere which in turn will be
electric field. As the bare mass --> 0, the acceleration increases causing the retarded field to have an increasing effect on the charge distribution, noting that I only have to worry about the retarded electric field for steady state conditions in theWhen we discussed what could "intuitively" be expected, I onlyAgreeing with you that the lowest energy will eventually beAlso here the solution will not be unique.. Especially if you release
What is the charge distribution of a hyperbolically accelerated charged hollow sphere?
the sphere quite suddenly. Then there certainly will be a "sloshing >>>>>> mode" for the charge in the early stages.
Intuitively, I'd expect this retarded self E' to lower the self energy of the polarized static sphere by depolarizing it, thus providing the necessary energy to redistribute the charge: perhaps the charge is evenly redistributed?I would expect the same (after some steady state is reached) but I'm
not sure if this is easy to prove..
You're agreeing with me here,
reached, also agreeing that depolarization will happen after
initial polarization, but it will not be depolarized back
to zero. After the E-field is (magically) switched on and the
sphere starts to accelerate, I would (intuitively) expect the
polarization vs. time to be something like this:
* *
* * * * * *
* * *
* * *
0*
Of course the 'sloshing' will consist of infinitely many
oscillations, decreasing exponentially in strength.
in that the charge will end up evenly distributed in the steady state when the sphere is hyperbolically accelerated: the applied E field polarizing the charge and the retarded E' field depolarizing it, cancelling one another exactly.It isn't what I would intuitively expect, I'd expect that as
long as the acceleration lasts, the charge is being pulled a
little bit ahead of the sphere's centre, so it remains
polarized all the time.
But doesn't this imply the charge is also uniformly distributed in the equivalent case with a gravitational field replacing the accelerated frame?My statement would imply that the sphere is *also polarized*
when hovering in the gravitational field. Also there, the
charge is pulled upwards by the E-field and will be off-center.
And in the latter case I am much more confident that this is
easy to prove! (Of course we then do have to prove in addition
that the equivalence principle actually applies here, but still
this reduces the original problem to proving two sub-problems.)
--
Jos
I agree with the above for a sphere with a finite bare mass that dominates the electromagnetic mass. And I also agree with what you've written elsewhere in this thread:
https://groups.google.com/g/sci.physics.electromag/c/JTPBOZ3W4wU/m/bqoSBw5fCQAJ
"The equivalence principle will not change the charge. It may deform >>> the sphere by Lorentz contraction when going from one frame to the
other, but in thic case that won't change a non-uniform distribution >>> into a uniform one.
And the charge will be non-uniform. Except in the case when the sphere >>> has zero mass (and all mass is in the charged particles and in the
E-field density, and no mass in the rigid sphere. But that is
unphysical)."
Would I be correct in saying that you believe the polarizing and depolarizing effects on the surface charge approach cancelling one another as the electromagnetic mass >> bare mass?
considered decent, simple, 19th-century electrostatics, with
physical objects that exist in everyday life!
With a mass-zero sphere (and mass-less charge carriers as well)
things might start to escape our range of intuition.. I would
not be sure whether the pull of gravity will actually change the
direction of field lines, for instance, if the system is hovering
in a constant gravity field. The problem is that, even if the
equivalence principle allows us to use that frame, I do not know
how Maxwell's equations might change in a non-inertial frame. (It
probably can be looked up on Wikipedia, but intuitively I would
just expect the unexpected.)
For safety, I would go back to the original frame, where we know
Maxwell, and try to solve it there. But the combination of E-
and B-fields and moving charges make this complicated of course.
The precise shape of the field is not obvious to begin with, and
since you ask whether some things might be precisely "cancelled",
you would really need precise information!
Initially, I can imagine the sphere to have a large bare mass compared to the electromagnetic so that it's 'gently" accelerated. The surface will obviously be an equipotential with a non-uniform charge distribution giving rise to an internal 0
although to be sure I need to calculate it of course.For hyperbolic motion, the LAD equation for a point charge has the Shott and radiation terms cancelling one another; the Shott term being the "acceleration energy" above. So likewise I'm confident of this cancelling for a charge sphere also,
Alternatively, we can work in the sphere's own stationary frame
and *assume* that Maxwell is unaltered for electrostatics in a
gravity field. In that case your proposed solution with uniform
charge does not seem to be stable. There is an upward pull of the
E-field on the charge carriers (which have no mass so they are
not pulled down by gravity) and there must be a downward pull of
gravity on the E-field energy density around the sphere (which
contains all the mass now). So intuitively we might still expect
the charge to move off centre (upwards) and the E-field lines to
be drooping downwards.. but the latter is incompatible with the
starting assumption that electrostatics is unaltered by gravity.
(So probably it isn't!)
GR looks like a minefield to me even for the experts which is why I try to keep away from it.But that makes it intriguing to see which mines there are in any
given case! In the case here it is the EM-field's behavior in a non- inertial frame, i.e. with constant gravity present. In that case
Maxwell's equations, and even simple Coulomb electrostatics, do not
apply unaltered!
As you can read in Wikipedia's Rohrlich paragraph below, "Maxwell equations do not hold" and a bit further "the gravitational field
slightly distorts the Coulomb field".
Of course we also know that combined E- and B-fields, like a moving
wave packet, would not propagate according to plain Maxwell, because
that would be rectilinear motion of the packet, and we know that
light is bent downwards by gravity. But it seems even electrostatics
is affected..
...All this does not answer your question whether the charge will be uniformly distributed in the sphere's (non-inertial) frame, but the
Resolution by Rohrlich https://en.wikipedia.org/wiki/Paradox_of_radiation_of_charged_particles_in_a_gravitational_field#Resolution_by_Rohrlich
fact that gravity "slightly distorts" the ordinary electrostatic situation, as stated, would make me believe (intuitively) that the
charge is not uniform any more..
--The Figure at the end of section three in this paper shows the fields in the instantaneous rest frame of a hyperbolically accelerated charged spherical shell:
Jos
The fields and self-force of a constantly accelerating spherical shell https://royalsocietypublishing.org/doi/10.1098/rspa.2013.0480
Do you agree that the direction of the electric field inside shown here is incorrect and needs to be reversed?
JMcA
On Tuesday, March 9, 2021 at 2:06:47 AM UTC, john mcandrew wrote:effected by the retarded self-field E' acting back on it etc. Which leads me to my question:
On Saturday, January 23, 2021 at 10:30:02 PM UTC, Jos Bergervoet wrote:
On 21/01/21 12:35 AM, john mcandrew wrote:
On Sunday, January 17, 2021 at 10:50:02 AM UTC, Jos Bergervoet wrote: >>>>> On 21/01/15 1:25 AM, john mcandrew wrote:
On Thursday, January 14, 2021 at 9:20:16 PM UTC, Jos Bergervoet wrote: >>>>>>> On 21/01/14 12:26 AM, john mcandrew wrote:
On Friday, December 25, 2020 at 11:08:29 AM UTC, Jos Bergervoet wrote: >>>>>>>>> On 20/12/23 5:06 PM, john mcandrew wrote:
Merry Christmas John, and to everyone else in the newsgroup! >>>>>>>>>> Let a continuous total charge +q be constrained to move freely on a spherical surface, referring to this from now on as a hollow charged sphere.
First, a Merry Christmas to the Christians and a celebratory solstice to everyone else! Now onto my question, and thanks in advance for any replies:
You have to give more information, especially about the mass (we could
have 3 mass contributions here: the mass of the supporting sphere, that
of the moving charge, and the electromagnetic mass of the surrounding >>>>>>>>> fields).
If this is statically placed in a constant electric field E, the surface charge will be pushed towards one end so that the total electric field inside returns to zero.For that case there is an exact solution (e.g. in Jackson). But that >>>>>>>>> is strictly speaking not unique. On top of it the charge can be >>>>>>>>> sloshing back and forward (of course these oscillations will die out >>>>>>>>> eventually, but still, the steady state will then only be reached >>>>>>>>> at t=infinity)
some time thes
Letting the sphere accelerate, I now have to take into account the self-field being retarded,But do you suddenly release the sphere? Or do you arrange for a more >>>>>>>>> "smooth" onset of the acelleration?
.. affecting each charge element in the rest frame of the sphere by a retarded electric field E'. This looks pretty difficult to calculate because I need to know the distribution of the polarized charge around the sphere which in turn will be
electric field. As the bare mass --> 0, the acceleration increases causing the retarded field to have an increasing effect on the charge distribution, noting that I only have to worry about the retarded electric field for steady state conditions in theWhen we discussed what could "intuitively" be expected, I onlyAgreeing with you that the lowest energy will eventually beAlso here the solution will not be unique.. Especially if you release >>>>>>>>> the sphere quite suddenly. Then there certainly will be a "sloshing >>>>>>>>> mode" for the charge in the early stages.
What is the charge distribution of a hyperbolically accelerated charged hollow sphere?
Intuitively, I'd expect this retarded self E' to lower the self energy of the polarized static sphere by depolarizing it, thus providing the necessary energy to redistribute the charge: perhaps the charge is evenly redistributed?I would expect the same (after some steady state is reached) but I'm >>>>>>>>> not sure if this is easy to prove..
You're agreeing with me here,
reached, also agreeing that depolarization will happen after
initial polarization, but it will not be depolarized back
to zero. After the E-field is (magically) switched on and the
sphere starts to accelerate, I would (intuitively) expect the
polarization vs. time to be something like this:
* *
* * * * * *
* * *
* * *
0*
Of course the 'sloshing' will consist of infinitely many
oscillations, decreasing exponentially in strength.
in that the charge will end up evenly distributed in the steady state when the sphere is hyperbolically accelerated: the applied E field polarizing the charge and the retarded E' field depolarizing it, cancelling one another exactly.It isn't what I would intuitively expect, I'd expect that as
long as the acceleration lasts, the charge is being pulled a
little bit ahead of the sphere's centre, so it remains
polarized all the time.
But doesn't this imply the charge is also uniformly distributed in the equivalent case with a gravitational field replacing the accelerated frame?My statement would imply that the sphere is *also polarized*
when hovering in the gravitational field. Also there, the
charge is pulled upwards by the E-field and will be off-center.
And in the latter case I am much more confident that this is
easy to prove! (Of course we then do have to prove in addition
that the equivalence principle actually applies here, but still
this reduces the original problem to proving two sub-problems.)
--
Jos
I agree with the above for a sphere with a finite bare mass that dominates the electromagnetic mass. And I also agree with what you've written elsewhere in this thread:
https://groups.google.com/g/sci.physics.electromag/c/JTPBOZ3W4wU/m/bqoSBw5fCQAJ
"The equivalence principle will not change the charge. It may deform >>>>>> the sphere by Lorentz contraction when going from one frame to the >>>>>> other, but in thic case that won't change a non-uniform distribution >>>>>> into a uniform one.
And the charge will be non-uniform. Except in the case when the sphere >>>>>> has zero mass (and all mass is in the charged particles and in the >>>>>> E-field density, and no mass in the rigid sphere. But that is
unphysical)."
Would I be correct in saying that you believe the polarizing and depolarizing effects on the surface charge approach cancelling one another as the electromagnetic mass >> bare mass?
considered decent, simple, 19th-century electrostatics, with
physical objects that exist in everyday life!
With a mass-zero sphere (and mass-less charge carriers as well)
things might start to escape our range of intuition.. I would
not be sure whether the pull of gravity will actually change the
direction of field lines, for instance, if the system is hovering
in a constant gravity field. The problem is that, even if the
equivalence principle allows us to use that frame, I do not know
how Maxwell's equations might change in a non-inertial frame. (It
probably can be looked up on Wikipedia, but intuitively I would
just expect the unexpected.)
For safety, I would go back to the original frame, where we know
Maxwell, and try to solve it there. But the combination of E-
and B-fields and moving charges make this complicated of course.
The precise shape of the field is not obvious to begin with, and
since you ask whether some things might be precisely "cancelled",
you would really need precise information!
Initially, I can imagine the sphere to have a large bare mass compared to the electromagnetic so that it's 'gently" accelerated. The surface will obviously be an equipotential with a non-uniform charge distribution giving rise to an internal 0
although to be sure I need to calculate it of course.
For hyperbolic motion, the LAD equation for a point charge has the Shott and radiation terms cancelling one another; the Shott term being the "acceleration energy" above. So likewise I'm confident of this cancelling for a charge sphere also,
The Figure at the end of section three in this paper shows the fields in the instantaneous rest frame of a hyperbolically accelerated charged spherical shell:But that makes it intriguing to see which mines there are in any
Alternatively, we can work in the sphere's own stationary frame
and *assume* that Maxwell is unaltered for electrostatics in a
gravity field. In that case your proposed solution with uniform
charge does not seem to be stable. There is an upward pull of the
E-field on the charge carriers (which have no mass so they are
not pulled down by gravity) and there must be a downward pull of
gravity on the E-field energy density around the sphere (which
contains all the mass now). So intuitively we might still expect
the charge to move off centre (upwards) and the E-field lines to
be drooping downwards.. but the latter is incompatible with the
starting assumption that electrostatics is unaltered by gravity.
(So probably it isn't!)
GR looks like a minefield to me even for the experts which is why I try to keep away from it.
given case! In the case here it is the EM-field's behavior in a non-
inertial frame, i.e. with constant gravity present. In that case
Maxwell's equations, and even simple Coulomb electrostatics, do not
apply unaltered!
As you can read in Wikipedia's Rohrlich paragraph below, "Maxwell
equations do not hold" and a bit further "the gravitational field
slightly distorts the Coulomb field".
Of course we also know that combined E- and B-fields, like a moving
wave packet, would not propagate according to plain Maxwell, because
that would be rectilinear motion of the packet, and we know that
light is bent downwards by gravity. But it seems even electrostatics
is affected..
...All this does not answer your question whether the charge will be
Resolution by Rohrlich
https://en.wikipedia.org/wiki/Paradox_of_radiation_of_charged_particles_in_a_gravitational_field#Resolution_by_Rohrlich
uniformly distributed in the sphere's (non-inertial) frame, but the
fact that gravity "slightly distorts" the ordinary electrostatic
situation, as stated, would make me believe (intuitively) that the
charge is not uniform any more..
--
Jos
The fields and self-force of a constantly accelerating spherical shell
https://royalsocietypublishing.org/doi/10.1098/rspa.2013.0480
Do you agree that the direction of the electric field inside shown here is incorrect and needs to be reversed?
JMcA
On second thoughts it looks right: its internal electric field cancels the applied electric field along the +x-axis giving a zero net electric field inside.
On 21/03/10 12:30 AM, john mcandrew wrote:be effected by the retarded self-field E' acting back on it etc. Which leads me to my question:
On Tuesday, March 9, 2021 at 2:06:47 AM UTC, john mcandrew wrote:
On Saturday, January 23, 2021 at 10:30:02 PM UTC, Jos Bergervoet wrote: >>> On 21/01/21 12:35 AM, john mcandrew wrote:
On Sunday, January 17, 2021 at 10:50:02 AM UTC, Jos Bergervoet wrote: >>>>> On 21/01/15 1:25 AM, john mcandrew wrote:
On Thursday, January 14, 2021 at 9:20:16 PM UTC, Jos Bergervoet wrote:
On 21/01/14 12:26 AM, john mcandrew wrote:
On Friday, December 25, 2020 at 11:08:29 AM UTC, Jos Bergervoet wrote:
On 20/12/23 5:06 PM, john mcandrew wrote:
Merry Christmas John, and to everyone else in the newsgroup! >>>>>>>>>> Let a continuous total charge +q be constrained to move freely on a spherical surface, referring to this from now on as a hollow charged sphere.
First, a Merry Christmas to the Christians and a celebratory solstice to everyone else! Now onto my question, and thanks in advance for any replies:
You have to give more information, especially about the mass (we could
have 3 mass contributions here: the mass of the supporting sphere, that
of the moving charge, and the electromagnetic mass of the surrounding
fields).
If this is statically placed in a constant electric field E, the surface charge will be pushed towards one end so that the total electric field inside returns to zero.For that case there is an exact solution (e.g. in Jackson). But that
is strictly speaking not unique. On top of it the charge can be >>>>>>>>> sloshing back and forward (of course these oscillations will die out
eventually, but still, the steady state will then only be reached >>>>>>>>> at t=infinity)
some time thes
Letting the sphere accelerate, I now have to take into account the self-field being retarded,But do you suddenly release the sphere? Or do you arrange for a more
"smooth" onset of the acelleration?
.. affecting each charge element in the rest frame of the sphere by a retarded electric field E'. This looks pretty difficult to calculate because I need to know the distribution of the polarized charge around the sphere which in turn will
electric field. As the bare mass --> 0, the acceleration increases causing the retarded field to have an increasing effect on the charge distribution, noting that I only have to worry about the retarded electric field for steady state conditions in theWhen we discussed what could "intuitively" be expected, I onlyAgreeing with you that the lowest energy will eventually beAlso here the solution will not be unique.. Especially if you release
What is the charge distribution of a hyperbolically accelerated charged hollow sphere?
the sphere quite suddenly. Then there certainly will be a "sloshing
mode" for the charge in the early stages.
Intuitively, I'd expect this retarded self E' to lower the self energy of the polarized static sphere by depolarizing it, thus providing the necessary energy to redistribute the charge: perhaps the charge is evenly redistributed?I would expect the same (after some steady state is reached) but I'm
not sure if this is easy to prove..
You're agreeing with me here,
reached, also agreeing that depolarization will happen after
initial polarization, but it will not be depolarized back
to zero. After the E-field is (magically) switched on and the >>>>>>> sphere starts to accelerate, I would (intuitively) expect the >>>>>>> polarization vs. time to be something like this:
* *
* * * * * *
* * *
* * *
0*
Of course the 'sloshing' will consist of infinitely many
oscillations, decreasing exponentially in strength.
in that the charge will end up evenly distributed in the steady state when the sphere is hyperbolically accelerated: the applied E field polarizing the charge and the retarded E' field depolarizing it, cancelling one another exactly.It isn't what I would intuitively expect, I'd expect that as
long as the acceleration lasts, the charge is being pulled a
little bit ahead of the sphere's centre, so it remains
polarized all the time.
But doesn't this imply the charge is also uniformly distributed in the equivalent case with a gravitational field replacing the accelerated frame?My statement would imply that the sphere is *also polarized*
when hovering in the gravitational field. Also there, the
charge is pulled upwards by the E-field and will be off-center. >>>>>>>
And in the latter case I am much more confident that this is
easy to prove! (Of course we then do have to prove in addition >>>>>>> that the equivalence principle actually applies here, but still >>>>>>> this reduces the original problem to proving two sub-problems.) >>>>>>>
--
Jos
I agree with the above for a sphere with a finite bare mass that dominates the electromagnetic mass. And I also agree with what you've written elsewhere in this thread:
https://groups.google.com/g/sci.physics.electromag/c/JTPBOZ3W4wU/m/bqoSBw5fCQAJ
"The equivalence principle will not change the charge. It may deform >>>>>> the sphere by Lorentz contraction when going from one frame to the >>>>>> other, but in thic case that won't change a non-uniform distribution >>>>>> into a uniform one.
And the charge will be non-uniform. Except in the case when the sphere
has zero mass (and all mass is in the charged particles and in the >>>>>> E-field density, and no mass in the rigid sphere. But that is
unphysical)."
Would I be correct in saying that you believe the polarizing and depolarizing effects on the surface charge approach cancelling one another as the electromagnetic mass >> bare mass?
considered decent, simple, 19th-century electrostatics, with
physical objects that exist in everyday life!
With a mass-zero sphere (and mass-less charge carriers as well)
things might start to escape our range of intuition.. I would
not be sure whether the pull of gravity will actually change the
direction of field lines, for instance, if the system is hovering >>>>> in a constant gravity field. The problem is that, even if the
equivalence principle allows us to use that frame, I do not know
how Maxwell's equations might change in a non-inertial frame. (It >>>>> probably can be looked up on Wikipedia, but intuitively I would
just expect the unexpected.)
For safety, I would go back to the original frame, where we know
Maxwell, and try to solve it there. But the combination of E-
and B-fields and moving charges make this complicated of course.
The precise shape of the field is not obvious to begin with, and
since you ask whether some things might be precisely "cancelled", >>>>> you would really need precise information!
Initially, I can imagine the sphere to have a large bare mass compared to the electromagnetic so that it's 'gently" accelerated. The surface will obviously be an equipotential with a non-uniform charge distribution giving rise to an internal 0
although to be sure I need to calculate it of course.
For hyperbolic motion, the LAD equation for a point charge has the Shott and radiation terms cancelling one another; the Shott term being the "acceleration energy" above. So likewise I'm confident of this cancelling for a charge sphere also,
The Figure at the end of section three in this paper shows the fields in the instantaneous rest frame of a hyperbolically accelerated charged spherical shell:But that makes it intriguing to see which mines there are in any
Alternatively, we can work in the sphere's own stationary frame
and *assume* that Maxwell is unaltered for electrostatics in a
gravity field. In that case your proposed solution with uniform
charge does not seem to be stable. There is an upward pull of the >>>>> E-field on the charge carriers (which have no mass so they are
not pulled down by gravity) and there must be a downward pull of
gravity on the E-field energy density around the sphere (which
contains all the mass now). So intuitively we might still expect
the charge to move off centre (upwards) and the E-field lines to
be drooping downwards.. but the latter is incompatible with the
starting assumption that electrostatics is unaltered by gravity.
(So probably it isn't!)
GR looks like a minefield to me even for the experts which is why I try to keep away from it.
given case! In the case here it is the EM-field's behavior in a non-
inertial frame, i.e. with constant gravity present. In that case
Maxwell's equations, and even simple Coulomb electrostatics, do not
apply unaltered!
As you can read in Wikipedia's Rohrlich paragraph below, "Maxwell
equations do not hold" and a bit further "the gravitational field
slightly distorts the Coulomb field".
Of course we also know that combined E- and B-fields, like a moving
wave packet, would not propagate according to plain Maxwell, because
that would be rectilinear motion of the packet, and we know that
light is bent downwards by gravity. But it seems even electrostatics
is affected..
...All this does not answer your question whether the charge will be
Resolution by Rohrlich
https://en.wikipedia.org/wiki/Paradox_of_radiation_of_charged_particles_in_a_gravitational_field#Resolution_by_Rohrlich
uniformly distributed in the sphere's (non-inertial) frame, but the
fact that gravity "slightly distorts" the ordinary electrostatic
situation, as stated, would make me believe (intuitively) that the
charge is not uniform any more..
--
Jos
The fields and self-force of a constantly accelerating spherical shell
https://royalsocietypublishing.org/doi/10.1098/rspa.2013.0480
Do you agree that the direction of the electric field inside shown here is incorrect and needs to be reversed?
JMcA
On second thoughts it looks right: its internal electric field cancels the applied electric field along the +x-axis giving a zero net electric field inside.Thanks for theinteresting reference! What is a bit strange, however, is
that the abstract says about their result "this is conjectured to be
exact", whereas the introduction describes the approach as "restricting
the motion to a simple case, and solving it exactly." So which is it?
What is very pleasing to see is that the author does actually define
what is meant by the rigid motion with constant acceleration, which
would of course otherwise lend itself for dozens of interpretations.
Here it is "moves such that there exists a frame in which the whole
shell is at rest at some moment", period.
As for the drawing of the field lines that you mention: I believe there
is no "applied" electric field! I did not see it mentioned in the text before that point. I think the assumption is that the acceleration is
caused by other forces, left unspecified, and that the calculation
only looks at what field is then created by the shell itself.
And whether the drawing is correct I can't judge, I did not try to
follow the details of the calculation. But there's nothing in the
picture that looks impossible. The field looks divergence-free in
empty space, and you can always find a charge distribution on the
shell that gives such a field (starting with a double layer for the
case of a Faraday cage with external and internal surface charges
to do the two seperate jobs of creating those fields, and then just
uniting them to one layer.)
What worries me a little is that those field lines do not have zero tangential component (along the shell surface). So why wouldn't the
charges move? It is a conducting sphere, I'd expect. So maybe you
are right and the drawing is at least a bit sloppy..
On Saturday, March 13, 2021 at 6:30:03 PM UTC, Jos Bergervoet wrote:be effected by the retarded self-field E' acting back on it etc. Which leads me to my question:
On 21/03/10 12:30 AM, john mcandrew wrote:
On Tuesday, March 9, 2021 at 2:06:47 AM UTC, john mcandrew wrote:
On Saturday, January 23, 2021 at 10:30:02 PM UTC, Jos Bergervoet wrote: >>> On 21/01/21 12:35 AM, john mcandrew wrote:
On Sunday, January 17, 2021 at 10:50:02 AM UTC, Jos Bergervoet wrote: >>>>> On 21/01/15 1:25 AM, john mcandrew wrote:
On Thursday, January 14, 2021 at 9:20:16 PM UTC, Jos Bergervoet wrote:
On 21/01/14 12:26 AM, john mcandrew wrote:
On Friday, December 25, 2020 at 11:08:29 AM UTC, Jos Bergervoet wrote:
On 20/12/23 5:06 PM, john mcandrew wrote:
Merry Christmas John, and to everyone else in the newsgroup! >>>>>>>>>> Let a continuous total charge +q be constrained to move freely on a spherical surface, referring to this from now on as a hollow charged sphere.
First, a Merry Christmas to the Christians and a celebratory solstice to everyone else! Now onto my question, and thanks in advance for any replies:
You have to give more information, especially about the mass (we could
have 3 mass contributions here: the mass of the supporting sphere, that
of the moving charge, and the electromagnetic mass of the surrounding
fields).
If this is statically placed in a constant electric field E, the surface charge will be pushed towards one end so that the total electric field inside returns to zero.For that case there is an exact solution (e.g. in Jackson). But that
is strictly speaking not unique. On top of it the charge can be >>>>>>>>> sloshing back and forward (of course these oscillations will die out
eventually, but still, the steady state will then only be reached
at t=infinity)
some time thes
Letting the sphere accelerate, I now have to take into account the self-field being retarded,But do you suddenly release the sphere? Or do you arrange for a more
"smooth" onset of the acelleration?
.. affecting each charge element in the rest frame of the sphere by a retarded electric field E'. This looks pretty difficult to calculate because I need to know the distribution of the polarized charge around the sphere which in turn will
electric field. As the bare mass --> 0, the acceleration increases causing the retarded field to have an increasing effect on the charge distribution, noting that I only have to worry about the retarded electric field for steady state conditions in theWhen we discussed what could "intuitively" be expected, I onlyAgreeing with you that the lowest energy will eventually be >>>>>>> reached, also agreeing that depolarization will happen after >>>>>>> initial polarization, but it will not be depolarized backAlso here the solution will not be unique.. Especially if you release
What is the charge distribution of a hyperbolically accelerated charged hollow sphere?
the sphere quite suddenly. Then there certainly will be a "sloshing
mode" for the charge in the early stages.
Intuitively, I'd expect this retarded self E' to lower the self energy of the polarized static sphere by depolarizing it, thus providing the necessary energy to redistribute the charge: perhaps the charge is evenly redistributed?I would expect the same (after some steady state is reached) but I'm
not sure if this is easy to prove..
You're agreeing with me here,
to zero. After the E-field is (magically) switched on and the >>>>>>> sphere starts to accelerate, I would (intuitively) expect the >>>>>>> polarization vs. time to be something like this:
* *
* * * * * *
* * *
* * *
0*
Of course the 'sloshing' will consist of infinitely many
oscillations, decreasing exponentially in strength.
in that the charge will end up evenly distributed in the steady state when the sphere is hyperbolically accelerated: the applied E field polarizing the charge and the retarded E' field depolarizing it, cancelling one another exactly.It isn't what I would intuitively expect, I'd expect that as >>>>>>> long as the acceleration lasts, the charge is being pulled a >>>>>>> little bit ahead of the sphere's centre, so it remains
polarized all the time.
But doesn't this imply the charge is also uniformly distributed in the equivalent case with a gravitational field replacing the accelerated frame?My statement would imply that the sphere is *also polarized* >>>>>>> when hovering in the gravitational field. Also there, the
charge is pulled upwards by the E-field and will be off-center. >>>>>>>
And in the latter case I am much more confident that this is >>>>>>> easy to prove! (Of course we then do have to prove in addition >>>>>>> that the equivalence principle actually applies here, but still >>>>>>> this reduces the original problem to proving two sub-problems.) >>>>>>>
--
Jos
I agree with the above for a sphere with a finite bare mass that dominates the electromagnetic mass. And I also agree with what you've written elsewhere in this thread:
https://groups.google.com/g/sci.physics.electromag/c/JTPBOZ3W4wU/m/bqoSBw5fCQAJ
"The equivalence principle will not change the charge. It may deform
the sphere by Lorentz contraction when going from one frame to the >>>>>> other, but in thic case that won't change a non-uniform distribution
into a uniform one.
And the charge will be non-uniform. Except in the case when the sphere
has zero mass (and all mass is in the charged particles and in the >>>>>> E-field density, and no mass in the rigid sphere. But that is >>>>>> unphysical)."
Would I be correct in saying that you believe the polarizing and depolarizing effects on the surface charge approach cancelling one another as the electromagnetic mass >> bare mass?
considered decent, simple, 19th-century electrostatics, with
physical objects that exist in everyday life!
With a mass-zero sphere (and mass-less charge carriers as well) >>>>> things might start to escape our range of intuition.. I would
not be sure whether the pull of gravity will actually change the >>>>> direction of field lines, for instance, if the system is hovering >>>>> in a constant gravity field. The problem is that, even if the
equivalence principle allows us to use that frame, I do not know >>>>> how Maxwell's equations might change in a non-inertial frame. (It >>>>> probably can be looked up on Wikipedia, but intuitively I would >>>>> just expect the unexpected.)
For safety, I would go back to the original frame, where we know >>>>> Maxwell, and try to solve it there. But the combination of E-
and B-fields and moving charges make this complicated of course. >>>>> The precise shape of the field is not obvious to begin with, and >>>>> since you ask whether some things might be precisely "cancelled", >>>>> you would really need precise information!
Initially, I can imagine the sphere to have a large bare mass compared to the electromagnetic so that it's 'gently" accelerated. The surface will obviously be an equipotential with a non-uniform charge distribution giving rise to an internal 0
although to be sure I need to calculate it of course.
For hyperbolic motion, the LAD equation for a point charge has the Shott and radiation terms cancelling one another; the Shott term being the "acceleration energy" above. So likewise I'm confident of this cancelling for a charge sphere also,
The Figure at the end of section three in this paper shows the fields in the instantaneous rest frame of a hyperbolically accelerated charged spherical shell:But that makes it intriguing to see which mines there are in any
Alternatively, we can work in the sphere's own stationary frame >>>>> and *assume* that Maxwell is unaltered for electrostatics in a
gravity field. In that case your proposed solution with uniform >>>>> charge does not seem to be stable. There is an upward pull of the >>>>> E-field on the charge carriers (which have no mass so they are
not pulled down by gravity) and there must be a downward pull of >>>>> gravity on the E-field energy density around the sphere (which
contains all the mass now). So intuitively we might still expect >>>>> the charge to move off centre (upwards) and the E-field lines to >>>>> be drooping downwards.. but the latter is incompatible with the >>>>> starting assumption that electrostatics is unaltered by gravity. >>>>> (So probably it isn't!)
GR looks like a minefield to me even for the experts which is why I try to keep away from it.
given case! In the case here it is the EM-field's behavior in a non- >>> inertial frame, i.e. with constant gravity present. In that case
Maxwell's equations, and even simple Coulomb electrostatics, do not >>> apply unaltered!
As you can read in Wikipedia's Rohrlich paragraph below, "Maxwell
equations do not hold" and a bit further "the gravitational field
slightly distorts the Coulomb field".
Of course we also know that combined E- and B-fields, like a moving >>> wave packet, would not propagate according to plain Maxwell, because >>> that would be rectilinear motion of the packet, and we know that
light is bent downwards by gravity. But it seems even electrostatics >>> is affected..
...All this does not answer your question whether the charge will be
Resolution by Rohrlich
https://en.wikipedia.org/wiki/Paradox_of_radiation_of_charged_particles_in_a_gravitational_field#Resolution_by_Rohrlich
uniformly distributed in the sphere's (non-inertial) frame, but the >>> fact that gravity "slightly distorts" the ordinary electrostatic
situation, as stated, would make me believe (intuitively) that the
charge is not uniform any more..
--
Jos
The fields and self-force of a constantly accelerating spherical shell >> https://royalsocietypublishing.org/doi/10.1098/rspa.2013.0480
Do you agree that the direction of the electric field inside shown here is incorrect and needs to be reversed?
JMcA
On second thoughts it looks right: its internal electric field cancels the applied electric field along the +x-axis giving a zero net electric field inside.
What worries me a little is that those field lines do not have zero tangential component (along the shell surface). So why wouldn't the charges move? It is a conducting sphere, I'd expect. So maybe you
are right and the drawing is at least a bit sloppy..
The applied electric field E provides the additional necessary tangential component.
On Saturday, March 13, 2021 at 11:30:29 PM UTC, john mcandrew wrote:will be effected by the retarded self-field E' acting back on it etc. Which leads me to my question:
On Saturday, March 13, 2021 at 6:30:03 PM UTC, Jos Bergervoet wrote:
On 21/03/10 12:30 AM, john mcandrew wrote:
On Tuesday, March 9, 2021 at 2:06:47 AM UTC, john mcandrew wrote:
On Saturday, January 23, 2021 at 10:30:02 PM UTC, Jos Bergervoet wrote:
On 21/01/21 12:35 AM, john mcandrew wrote:
On Sunday, January 17, 2021 at 10:50:02 AM UTC, Jos Bergervoet wrote:
On 21/01/15 1:25 AM, john mcandrew wrote:
On Thursday, January 14, 2021 at 9:20:16 PM UTC, Jos Bergervoet wrote:
On 21/01/14 12:26 AM, john mcandrew wrote:
On Friday, December 25, 2020 at 11:08:29 AM UTC, Jos Bergervoet wrote:
On 20/12/23 5:06 PM, john mcandrew wrote:
Merry Christmas John, and to everyone else in the newsgroup! >>>>>>>>>> Let a continuous total charge +q be constrained to move freely on a spherical surface, referring to this from now on as a hollow charged sphere.
First, a Merry Christmas to the Christians and a celebratory solstice to everyone else! Now onto my question, and thanks in advance for any replies:
You have to give more information, especially about the mass (we could
have 3 mass contributions here: the mass of the supporting sphere, that
of the moving charge, and the electromagnetic mass of the surrounding
fields).
If this is statically placed in a constant electric field E, the surface charge will be pushed towards one end so that the total electric field inside returns to zero.For that case there is an exact solution (e.g. in Jackson). But that
is strictly speaking not unique. On top of it the charge can be
sloshing back and forward (of course these oscillations will die out
eventually, but still, the steady state will then only be reached
at t=infinity)
some time thes
Letting the sphere accelerate, I now have to take into account the self-field being retarded,But do you suddenly release the sphere? Or do you arrange for a more
"smooth" onset of the acelleration?
.. affecting each charge element in the rest frame of the sphere by a retarded electric field E'. This looks pretty difficult to calculate because I need to know the distribution of the polarized charge around the sphere which in turn
electric field. As the bare mass --> 0, the acceleration increases causing the retarded field to have an increasing effect on the charge distribution, noting that I only have to worry about the retarded electric field for steady state conditions in theWhen we discussed what could "intuitively" be expected, I only >>>>> considered decent, simple, 19th-century electrostatics, withAgreeing with you that the lowest energy will eventually be >>>>>>> reached, also agreeing that depolarization will happen after >>>>>>> initial polarization, but it will not be depolarized back >>>>>>> to zero. After the E-field is (magically) switched on and the >>>>>>> sphere starts to accelerate, I would (intuitively) expect the >>>>>>> polarization vs. time to be something like this:Also here the solution will not be unique.. Especially if you release
What is the charge distribution of a hyperbolically accelerated charged hollow sphere?
the sphere quite suddenly. Then there certainly will be a "sloshing
mode" for the charge in the early stages.
Intuitively, I'd expect this retarded self E' to lower the self energy of the polarized static sphere by depolarizing it, thus providing the necessary energy to redistribute the charge: perhaps the charge is evenly redistributed?I would expect the same (after some steady state is reached) but I'm
not sure if this is easy to prove..
You're agreeing with me here,
* *
* * * * * *
* * *
* * *
0*
Of course the 'sloshing' will consist of infinitely many
oscillations, decreasing exponentially in strength.
in that the charge will end up evenly distributed in the steady state when the sphere is hyperbolically accelerated: the applied E field polarizing the charge and the retarded E' field depolarizing it, cancelling one another exactly.It isn't what I would intuitively expect, I'd expect that as >>>>>>> long as the acceleration lasts, the charge is being pulled a >>>>>>> little bit ahead of the sphere's centre, so it remains
polarized all the time.
But doesn't this imply the charge is also uniformly distributed in the equivalent case with a gravitational field replacing the accelerated frame?My statement would imply that the sphere is *also polarized* >>>>>>> when hovering in the gravitational field. Also there, the >>>>>>> charge is pulled upwards by the E-field and will be off-center. >>>>>>>
And in the latter case I am much more confident that this is >>>>>>> easy to prove! (Of course we then do have to prove in addition >>>>>>> that the equivalence principle actually applies here, but still >>>>>>> this reduces the original problem to proving two sub-problems.) >>>>>>>
--
Jos
I agree with the above for a sphere with a finite bare mass that dominates the electromagnetic mass. And I also agree with what you've written elsewhere in this thread:
https://groups.google.com/g/sci.physics.electromag/c/JTPBOZ3W4wU/m/bqoSBw5fCQAJ
"The equivalence principle will not change the charge. It may deform
the sphere by Lorentz contraction when going from one frame to the
other, but in thic case that won't change a non-uniform distribution
into a uniform one.
And the charge will be non-uniform. Except in the case when the sphere
has zero mass (and all mass is in the charged particles and in the
E-field density, and no mass in the rigid sphere. But that is >>>>>> unphysical)."
Would I be correct in saying that you believe the polarizing and depolarizing effects on the surface charge approach cancelling one another as the electromagnetic mass >> bare mass?
physical objects that exist in everyday life!
With a mass-zero sphere (and mass-less charge carriers as well) >>>>> things might start to escape our range of intuition.. I would >>>>> not be sure whether the pull of gravity will actually change the >>>>> direction of field lines, for instance, if the system is hovering >>>>> in a constant gravity field. The problem is that, even if the >>>>> equivalence principle allows us to use that frame, I do not know >>>>> how Maxwell's equations might change in a non-inertial frame. (It >>>>> probably can be looked up on Wikipedia, but intuitively I would >>>>> just expect the unexpected.)
For safety, I would go back to the original frame, where we know >>>>> Maxwell, and try to solve it there. But the combination of E- >>>>> and B-fields and moving charges make this complicated of course. >>>>> The precise shape of the field is not obvious to begin with, and >>>>> since you ask whether some things might be precisely "cancelled", >>>>> you would really need precise information!
Initially, I can imagine the sphere to have a large bare mass compared to the electromagnetic so that it's 'gently" accelerated. The surface will obviously be an equipotential with a non-uniform charge distribution giving rise to an internal 0
although to be sure I need to calculate it of course.
For hyperbolic motion, the LAD equation for a point charge has the Shott and radiation terms cancelling one another; the Shott term being the "acceleration energy" above. So likewise I'm confident of this cancelling for a charge sphere also,
non EM force) not explicitly shown, but assumed in the diagram: this keeps the charges from moving wrt one another. Jos's comment has also made me rethink my OP where I claimed the retarded E field depolarizes the the charge so that it becomes continuous,The Figure at the end of section three in this paper shows the fields in the instantaneous rest frame of a hyperbolically accelerated charged spherical shell:But that makes it intriguing to see which mines there are in any
Alternatively, we can work in the sphere's own stationary frame >>>>> and *assume* that Maxwell is unaltered for electrostatics in a >>>>> gravity field. In that case your proposed solution with uniform >>>>> charge does not seem to be stable. There is an upward pull of the >>>>> E-field on the charge carriers (which have no mass so they are >>>>> not pulled down by gravity) and there must be a downward pull of >>>>> gravity on the E-field energy density around the sphere (which >>>>> contains all the mass now). So intuitively we might still expect >>>>> the charge to move off centre (upwards) and the E-field lines to >>>>> be drooping downwards.. but the latter is incompatible with the >>>>> starting assumption that electrostatics is unaltered by gravity. >>>>> (So probably it isn't!)
GR looks like a minefield to me even for the experts which is why I try to keep away from it.
given case! In the case here it is the EM-field's behavior in a non- >>> inertial frame, i.e. with constant gravity present. In that case
Maxwell's equations, and even simple Coulomb electrostatics, do not >>> apply unaltered!
As you can read in Wikipedia's Rohrlich paragraph below, "Maxwell >>> equations do not hold" and a bit further "the gravitational field >>> slightly distorts the Coulomb field".
Of course we also know that combined E- and B-fields, like a moving >>> wave packet, would not propagate according to plain Maxwell, because >>> that would be rectilinear motion of the packet, and we know that
light is bent downwards by gravity. But it seems even electrostatics >>> is affected..
...All this does not answer your question whether the charge will be >>> uniformly distributed in the sphere's (non-inertial) frame, but the >>> fact that gravity "slightly distorts" the ordinary electrostatic
Resolution by Rohrlich
https://en.wikipedia.org/wiki/Paradox_of_radiation_of_charged_particles_in_a_gravitational_field#Resolution_by_Rohrlich
situation, as stated, would make me believe (intuitively) that the >>> charge is not uniform any more..
--
Jos
The fields and self-force of a constantly accelerating spherical shell
https://royalsocietypublishing.org/doi/10.1098/rspa.2013.0480
Do you agree that the direction of the electric field inside shown here is incorrect and needs to be reversed?
JMcA
[snipped]On second thoughts it looks right: its internal electric field cancels the applied electric field along the +x-axis giving a zero net electric field inside.
What worries me a little is that those field lines do not have zero tangential component (along the shell surface). So why wouldn't the charges move? It is a conducting sphere, I'd expect. So maybe you
are right and the drawing is at least a bit sloppy..
The applied electric field E provides the additional necessary tangential component.I think I'm wrong here after rethinking about it. In the calculation the author assumes each component charge is accelerated by the necessary constant force keeping it rigid with its neighbours, and so comes under a force having a component (Poincare
Before I said yes, whereas now I find it embarrassingly obvious it's no, after realizing what's responsible for the redistribution of the charges and the electric field: the non EM forces of constraint keeping the charges confined to the surface of thesphere.
Despite these non-EM forces of constraint doing no net work on the charges, they're still able to move them around and hence change the distribution of the EM field while keeping its total energy constant.
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