# Equilibrium state of sourceless EM field

• I
• Creedence
In summary: So the equilibrium state is the sum of standing electromagnetic waves. And the amplitude of these waves are given by the Fourier series of B(t=0).In summary, a box made of perfectly conducting material has a localized magnetic field at some point in time. After this point, the magnetic field starts to spread and fill the box.
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

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

How do you get a "sourceless" magnetic field? $\nabla \cdot B = 0$.
No charges, no currents, and at t=0 no varying electric field.

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

So why is your field non-zero?

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.

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

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.

Last edited:
Creedence said:
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##

Dale said:
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.

Creedence said:
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.

Dale said:
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→∞ .

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.

## 1. What is the equilibrium state of a sourceless electromagnetic (EM) field?

The equilibrium state of a sourceless EM field is a state where there are no external sources present to generate or alter the field. This means that there are no charges or currents present in the field, and it is in a state of rest.

## 2. How is the equilibrium state of a sourceless EM field different from other states?

The equilibrium state of a sourceless EM field is different from other states because it does not require any external sources to maintain its state. In other states, such as a field generated by a charged particle, there is a continuous exchange of energy and momentum between the source and the field.

## 3. Can a sourceless EM field exist in a real-world scenario?

In theory, a sourceless EM field can exist in a real-world scenario. However, it is difficult to achieve in practice as there are always some external sources present, even if they are very small. In most cases, a sourceless EM field is used as a simplified model for theoretical calculations.

## 4. How is the equilibrium state of a sourceless EM field described mathematically?

The equilibrium state of a sourceless EM field is described mathematically by Maxwell's equations, which are a set of four partial differential equations that relate the electric and magnetic fields to their sources. In the case of a sourceless field, these equations reduce to the wave equation, which describes the propagation of electromagnetic waves.

## 5. What are some practical applications of understanding the equilibrium state of a sourceless EM field?

Understanding the equilibrium state of a sourceless EM field is important in many areas of science and technology. It is essential in the study of electromagnetic radiation, such as radio waves and light, and is used in the design of antennas and other communication devices. It also plays a crucial role in fields such as optics, electromagnetism, and quantum mechanics.

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