I Equilibrium state of sourceless EM field

[Creedence]

TL;DR Summary

What happends to a sourceless magnetic field in a confined space?

Given a box made of perfectly conducting material. At some part of it at t=0 there is a localized magnetic field. It's sourceless and there aren't any dissipation. After t=0 it starts to spread and fill the box. What is the equilibrium state?

Thanks for the answer(s),
Robert

 

[Vanadium 50]

How do you get a "sourceless" magnetic field? \nabla \cdot B = 0.

 

[Creedence]

How do you get a "sourceless" magnetic field? \nabla \cdot B = 0.

No charges, no currents, and at t=0 no varying electric field.

 

[Vanadium 50]

No charges, no currents, and at t=0 no varying electric field.

So why is your field non-zero?

 

[Creedence]

So why is your field non-zero?

Because I put it there as an initial condition and I'm interested in the system's time evolution.

 

[Vanadium 50]

Then this whole thread is "what do the laws of physics say when I violate the laws of physics".

 

[Creedence]

Then this whole thread is "what do the laws of physics say when I violate the laws of physics".

After I violated it. And because at t=0 the system is in a valid state, this question makes sense.

 

[Dale]

And because at t=0 the system is in a valid state

I don’t think that it is a valid state. How about this modification:

For $t < 0$ there is a small loop at the origin with a constant current which sets up a steady dipole magnetic field. At $t>0$ the current is 0. So there is a source, but it is switched off at $t=0$

 

[Creedence]

I don’t think that it is a valid state. How about this modification:

For $t < 0$ there is a small loop at the origin with a constant current which sets up a steady dipole magnetic field. At $t>0$ the current is 0. So there is a source, but it is switched off at $t=0$

The current change makes the initial field non-static. But I think its effect can be neglected. I'm only interested in the future of the built-up magnetic field.
Thanks, that is a good idea.

 

[Dale]

The current change makes the initial field non-static. But I think its effect can be neglected. I'm only interested in the future of the built-up magnetic field.
Thanks, that is a good idea.

So this is then fairly easy to analyze using Jefimenko’s equations. At each point you can divide time into two pieces: $t>t_r$ and $t<t_r$ where $t_r=(\sqrt{x^2+y^2+z^2})/c$ is the retarded time. Before $t_r$ the magnetic field will be the standard dipole field. After $t_r$ it will be 0. At $t_r$ there will be an impulsive E field which will satisfy Maxwell’s equations between the two conditions.

 

[Creedence]

So this is then fairly easy to analyze using Jefimenko’s equations. At each point you can divide time into two pieces: $t>t_r$ and $t<t_r$ where $t_r=(\sqrt{x^2+y^2+z^2})/c$ is the retarded time. Before $t_r$ the magnetic field will be the standard dipole field. After $t_r$ it will be 0. At $t_r$ there will be an impulsive E field which will satisfy Maxwell’s equations between the two conditions.

So the stationary magnetic field disappears (or radiates into the other parts of the box). The equilibrium state will be the sum of standing electromagnetic waves. And the amplitude of these waves are given by the Fourier series of B(t=0).
Is it OK?

I didn't count on the possible non-ergodicity of the system, so I assume it will reach a stationary state in t→∞ .

 

[Dale]

Oh, I forgot about the box. Since it is conductive you will get reflections. The field will be very complicated. I was just thinking of free space except for the small loop at the origin. That gives the simple fields I described above.