# How to setup $\vec E$ and $\vec B$ to allow for propagation of electromagnetic wavepacket? [closed]

When the two wave fields of $$\vec E$$ and $$\vec B$$ are given a priori, then the coupled equations

$$\frac{\partial \vec B}{\partial t} = curl (\vec E)$$

$$\frac{\partial \vec E}{\partial t} = - curl (\vec B)$$

completely determine the time evolution of $$\vec E$$ and $$\vec B$$ (where c=1, $$\mu_0$$=1, and j=0).

My questions are:

1. What is the standard "particle in a box" first test case problem for electromagnetic waves?

2. How to setup the initial $$\vec E$$ and $$\vec B$$ wave field at t=0 to allow for the propagation of an electromagnetic wavepacket / light?

• You mean in vacuum, right? Otherwise, you would have a source term in one of your equations...
– hft
Commented Dec 16, 2022 at 19:38
• Do you have an electromagnetism textbook to which you can refer? If not, I would suggest the one by Griffiths as a good undergraduate introduction.
– hft
Commented Dec 16, 2022 at 19:39
• @hft the $\vec E$ and $\vec B$ field by themselves are enough to completely specify their electromagnetic cross interaction, don't they? If there are sources present, they could have been internalized a priori into the initial distribution within the $\vec E$ and $\vec B$ field? Commented Dec 16, 2022 at 19:43
• Not generally... If there are sources present then there is a $J$ term in your second equation. But often we are interested in wave propagation through vacuum or a homogeneous medium where we can account for some effects, like polarization for example, via a linear dielectric function.
– hft
Commented Dec 16, 2022 at 19:47
• anyways, to proceed in vacuum, you can take the partial time derivative of your first equation and then plug in your second equation to get a wave equation. But... this is all textbook stuff, so it would be good to just crack a textbook...
– hft
Commented Dec 16, 2022 at 19:48

You can use the identity $$\nabla\times(\nabla\times f) = -\nabla^2 f + \nabla \cdot (\nabla\cdot f)$$ and the fact that $$\nabla\cdot B = 0$$.

Then you arrive at the wave equation for the electric and magnetic field: $$\nabla^2 E - \frac{\partial^2}{\partial t^2}E = 0$$

1. The easiest solution to this equation (which you call "particle in a box for electromagnetism", although I think this is more the analog to the free particle solution ;)) $$E(r,t) = E_0 e^{ik\cdot r}.$$

2. An illustrative example I think is a gaussian wavepacket that travels in z-direction with initial condition $$E(z,t=0) = e^{-z^2} e^{ikz}$$

In vacuum it will propagate with the group velocity $$v=c=1$$:

$$E(z,t) = e^{-(z-vt)^2} e^{ikz - i\omega t}.$$

Without dispersion it will always be the same gaussian wave packet, just translated in z-direction by $$vt$$.

You can find the corresponding magnetic field using Maxwell's equations.

• thank you vm! i am a little unclear about the complex exponential $\exp(ikz)$ in the solution. In electromagnetism, $\vec E$ and $\vec B$ are real valued, right? What components does the i separate? Commented Dec 16, 2022 at 20:00
• Yes, they are real-valued. Since the equations governing electromagnetism are linear, we can calculate them using complex-valued functions, which makes many things easier, and then taking either the real or the imaginary part of the formal solution to be the "real" solution. I just wonder, when you are already talking about particle in a box and stuff, shouldn't you then be familiar with basic electromagnetism? Anyway, as others pointed out, the first two chapters of a standard EM textbook should answer your questions. Commented Dec 16, 2022 at 20:09
• i will make sure to look it up. There are apparently a few solutions given in a lecture by feynman: feynmanlectures.caltech.edu/II_20.html Commented Dec 16, 2022 at 20:17