Solving for Magnetic Fields from time varying Electric fields

[GarageDweller]

Hi all, new to the forums.
Anyways I had a day off and decided to try something quirky with maxwells equations.
Using the Ampere-Maxwell equation, I tried to solve for the magnetic field that would be created by a <0,0,t> Electric field. (No current and no magnetic materials)

∇ x B = εμ(∂E/∂t)
∇ x B = εμ<0,0,1> (Pardon the bad notation)

∇ x B=< ∂Bz/∂y-∂By/∂z , ∂Bx/∂z-∂Bz/∂x , ∂By/∂x-∂Bx/∂y >

Equating the components, we have:

∂Bz/∂y=∂By/∂z

∂Bx/∂z=∂Bz/∂x

∂By/∂x-∂Bx/∂y=1

Here's the problem, How do I solve this mess?

 

[Muphrid]

That the source of the magnetic field the time-varying electric field and not an actual current makes no difference to Maxwell's equations. Because $\nabla \cdot B = 0$ still, you can remove the derivative through its inverse, the free-space Green's function $r/4\pi|r|^3$. What you get is

$$B(r) \propto \int_V \frac{\hat z \times (r - r')}{4 \pi |r - r'|^3} \; d^3r'$$

This result (when the proper proportionality constants are in) is the Biot-Savart law. The problem, however, is that if this electric field is time-varying everywhere, then I doubt this integral converges. You'll likely have to confine yourself to a 2D plane and invoke symmetry, or even simplify the source electric field to being nonzero only at a specific point, unless you're explicitly interested in the case of E being nonzero throughout a particular region.

 

[GarageDweller]

Sorry I'm not familiar with the method you're referring to here, could you elaborate?

 

[Muphrid]

You're not familiar with the use of Green's functions to solve differential equations? I'm not terribly surprised, I guess; the topic is sometimes treated like it's esoteric. Here's the basic idea:

Many differential equations are of the form $dA/dx = J$. It's first order, and it's simple. The Green's function for $d/dx$ is defined such that $dG/dx = \delta(x)$--the delta function.

This isn't immediately useful until you use it with the fundamental theorem of calculus, which can be written as

$$\left. A(x) \right|_a^b = \int_a^b \frac{dA}{dx} \; dx = \int_a^b J(x) \; dx$$

Now, using the fundamental theorem above (with $dA/dx = J$), just for fun, consider

$$\left. A(x') G(x-x') \right|_{x'=a}^b = \int_a^b \frac{dA(x')}{dx'} G(x-x') + A(x') \frac{dG(x-x')}{dx'} \; dx' = \int_a^b J(x') G(x-x') - A(x') \delta(x-x') \; dx'$$

Now, if we let $a \to -\infty$ and $b \to \infty$, we cover the whole real line. We generally require the solution for $G(x)$ to be well-behaved there, so the term on the left-hand side goes to zero. Remember also that when you integrate over a delta function, it forces the argument to zero. This leads to

$0 = \int_{-\infty}^\infty J(x') G(x-x') \; dx' - A(x) \implies A(x) = \int_{-\infty}^\infty J(x') G(x-x') \; dx'$

So, if you know the form of the Green's function for a particular differential operator and the source that generates the field, you can solve for the field through this integration. Extending this to 3D isn't too hard, but it requires some mathematical formalism that may obscure the point. The Green's function for $\nabla$ is well known--you use it every time you talk about the field from a point charge, because we model point charges as delta function sources anyway.

 

[blue_raver22]

Hello there, it's great to see someone exploring the applications of Maxwell's equations! Solving for magnetic fields from time-varying electric fields can be a challenging task, but it is an important part of understanding electromagnetic phenomena.

To solve this problem, you can use mathematical techniques such as vector calculus and partial differentiation. The first step would be to rewrite the equations in a more simplified form, using the notation provided. For example, you can write the Ampere-Maxwell equation as:

∂B/∂t = εμE

Then, you can use the equations for the components of the magnetic field (∂Bx/∂x, ∂By/∂y, ∂Bz/∂z) and substitute them into the equations for the components of the electric field (∂Ex/∂x, ∂Ey/∂y, ∂Ez/∂z). This will give you a system of equations that you can solve for each component of the magnetic field.

Alternatively, you can also use numerical methods to solve this problem. This involves discretizing the equations and using numerical techniques such as finite difference or finite element methods to approximate the solutions.

In both cases, it is important to keep in mind any boundary conditions or assumptions that may affect the solution. For example, the absence of current and magnetic materials in this problem may simplify the equations, but you should still consider the effects of these factors on the solution.

I hope this helps guide you in your exploration of Maxwell's equations. Keep up the good work!