Solving the EM field equations to produce the desired vector field

[greswd]

TL;DR Summary

I have a set of equations which represent conditions that the desired solution vector field has to meet.

So, we have A, the magnetic vector potential, and its divergence is the Lorenz gauge condition.

I want to solve for the two vector fields of F and G, and I'm wondering how I should begin$\nabla \cdot \mathbf{F}=-\nabla \cdot\frac{\partial}{\partial t}\mathbf{A} =-\frac{\partial}{\partial t}\left ( \nabla \cdot \mathbf{A} \right )$

$\nabla \times \mathbf{F} = 0$

$\nabla \cdot \mathbf{G} = 0$

$\nabla \times \mathbf{G} = \nabla \times\frac{\partial}{\partial t}\mathbf{A} =\frac{\partial}{\partial t}\left ( \nabla \times \mathbf{A} \right )$Also, solving for one of them, solving either F or G, is good enough

 

[Vanadium 50]

Probably numerically. And I noticed you never used the words "boundary conditions". They are important.

 

[greswd]

Probably numerically. And I noticed you never used the words "boundary conditions". They are important.

ahh yeah, for EM I guess it'd have to drop to zero at infinity

so you think an analytical solution might be out of reach?

 

[Vanadium 50]

For an arbitrary potential? For every single potential? Probably not.

 

[greswd]

For an arbitrary potential? For every single potential? Probably not.

does this mean like every single potential which is Lorenz gauged

 

[Delta2]

Ok I guess $\mathbf{A}$ is given and is known and you want to find $\mathbf{F}$ or $\mathbf{G}$ in terms of $\mathbf{A}$.

Have you heard of the fundamental theorem of vector calculus, also known as Helmholtz theorem?

If not, then use the substitution $$\mathbf{F}=\nabla F' (1)$$ then equation$\nabla\times\mathbf{F}=0$ is automatically satisfied (because the curl of the gradient is zero) and the second equation will lead you to Poisson's equation which you probably know how to solve.

Similarly for G, use the substitution $$\mathbf{G}=\nabla\times\mathbf{G'} (2)$$ and then the condition $\nabla\cdot G=0$ is satisfied (because the divergence of a curl is always zero) and the second equation will lead you to 3 Poisson's equations, one for each component of $\mathbf{G'}$. You will also need the condition that $\nabla\cdot\mathbf{G'}=0$ and the identity $\nabla\times(\nabla\times \mathbf{G'})=\nabla^2\mathbf{G'}-\nabla(\nabla\cdot\mathbf{G'})$

I am almost confident that you know the general solution to Poisson's equation (with the boundary condition that the field/potential is zero at infinity).

Once you find the scalar field $F'$ and the vector field $\mathbf{G'}$ as solutions to Poisson's, use (1) and (2) to compute $\mathbf{F}$ and $\mathbf{G}$.

 

[Delta2]

For an arbitrary potential? For every single potential? Probably not.

There is analytical solution in terms of an integral of the divergence or curl of A, but yes if A is really given and want to perform the integration you might need numerical methods.

 

[greswd]

There is analytical solution in terms of an integral of the divergence or curl of A, but yes if A is really given and want to perform the integration you might need numerical methods.

oh nice, i would like a general solution

 

[Delta2]

oh nice, i would like a general solution

Did you read my post #6? That is to define the auxiliary fields $F',\mathbf{G'}$ and solve for those first as solutions to Poisson's equation...