Maxwell equations with time-dependent boundary conditions

[checkfrogger]

Hi folks,

I was wondering how to code a Maxwell solver for a problem with time-dependent boundary conditions. This is not my homework, but I love programming and would like to implement what I learned in my physics undergrad course to get a better understanding.
More precisely, if I have an electrode with a time-dependent potential, how do I obtain the electric and magnetic field around it?
I basically came up with two ways, which both seem inappropriate to me:
1)
- calculate the potential using the Poisson equation with boundary conditions at time t=0
- then obtain E as the neg. gradient of the potential.
- calculate E and B at the next time step using the two curl equations of Maxwell equations
- repeat steps 1 and 2 at the next step and it might not be consistent with the third step...
I have the feeling that I mix electrostatics and electrodynamics here
2)
- set a boundary condition for E, solve the divergence equations of Maxwell equations at t=0
- calculate the next time step using the curl equations of Maxwell equations. The obtained E at t=1 might be inconsistent with the new potential at t=1.
-> similar problem here: I am not sure how to make this self-consistent

Thanks for your help!
Andrew

 

[Born2bwire]

Easiest way is the Finite Difference Time Domain (FDTD) analysis using the Yee algorithm which discretizes the differential equations in both time and position.

 

[checkfrogger]

Easiest way is the Finite Difference Time Domain (FDTD) analysis using the Yee algorithm which discretizes the differential equations in both time and position.

Thanks, but as far as I understand this algorithm only solves the two curl equations. It is fine if you start with a solution that satisfies the divergence equations.
My problem is still how to incorporate boundary conditions which are time-dependent and given in terms of the potential only.

 

[Born2bwire]

Many excitations in computational electromagnetics are given as voltages. The simplest is to do a delta gap source which is simple to do in FDTD as well. More complex methods would involve say finding the principle excitation mode of your source (like in a transmission line) and exciting the principle mode's field. Without having been given what the excitation is we can't really begin to provide any advice on how to model it. Still, any time or frequency domain computational solver like FDTD, finite element method (FEM) or method of moments (MOM) will probably be satisfactory for you. I would suggest looking at an appropriate text to see how excitations are handled. Taflove is good for FDTD and I like a recent text by Walter Gibson for MOM.

 

[sardar]

I can provide some insights on how to approach this problem. First of all, it is important to note that Maxwell's equations are fundamental laws of electromagnetism that describe the behavior of electric and magnetic fields. These equations are valid for both static and dynamic situations, which means they can be applied to problems with time-dependent boundary conditions.

To solve a problem with time-dependent boundary conditions using Maxwell's equations, one approach could be to use a numerical method such as finite difference or finite element methods. These methods involve discretizing the equations and solving them iteratively at each time step.

In terms of boundary conditions, it is important to specify both the electric and magnetic fields at the boundaries, as these are interdependent quantities. For example, if you have a time-varying potential on an electrode, you would need to specify the electric field at that boundary as well. This can be done by using the gradient of the potential, as you mentioned in your first approach.

Regarding your concerns about mixing electrostatics and electrodynamics, it is important to understand that these are two different regimes of Maxwell's equations. In electrostatics, the time derivatives of the fields are zero, while in electrodynamics they are non-zero. So, in your first approach, you are correctly using the Poisson equation (which is part of electrostatics) to calculate the potential and then using the curl equations (which are part of electrodynamics) to calculate the fields at the next time step.

For your second approach, setting a boundary condition for the electric field and solving the divergence equations at t=0 is a valid approach. However, to make it self-consistent, you would also need to specify the boundary conditions for the magnetic field and solve the curl equations at t=0. This will ensure that the fields are consistent with each other at the boundaries.

In summary, solving a problem with time-dependent boundary conditions using Maxwell's equations requires a careful consideration of both the equations and the boundary conditions. It is also important to choose an appropriate numerical method and ensure that the equations are solved self-consistently at each time step. I hope this helps in your coding project and further understanding of electromagnetism.