No charges, no currents, and at t=0 no varying electric field.Vanadium 50 said:How do you get a "sourceless" magnetic field? [itex]\nabla \cdot B = 0[/itex].
Creedence said:No charges, no currents, and at t=0 no varying electric field.
Because I put it there as an initial condition and I'm interested in the system's time evolution.Vanadium 50 said:So why is your field non-zero?
After I violated it. And because at t=0 the system is in a valid state, this question makes sense.Vanadium 50 said:Then this whole thread is "what do the laws of physics say when I violate the laws of physics".
I don’t think that it is a valid state. How about this modification:Creedence said:And because at t=0 the system is in a valid state
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.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##
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 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 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).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.
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.
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.
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.
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.
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.