Electromagnetic Waves

Spencer Boebel '24

Electromagnetic Waves

The Main Idea

An electromagnetic wave is what happens when we put all four of Maxwell's equations together. A changing electric field creates a magnetic field, which changes and creates an electric field, starting it all over again. This is the basic premise of an electric wave, and we shall derive other properties presently.

A Mathematical Model

The mathematical equations that allow us to model this topic are Maxwell's equations; but a simpler way to figure it out is to consider a notion from General Relativity. According to General Relativity, the speed of information is [math]\displaystyle{ c }[/math]. Now consider an electron moving at a velocity [math]\displaystyle{ \vec{v}(t) }[/math]. Now consider an observation point a location [math]\displaystyle{ \vec{r} }[/math] from the electron. Let us say the electron is oscillating, so we have that [math]\displaystyle{ \vec{v}=\cos{t} \vec{k} }[/math]. Then since the information that the electron is moving takes a while to get to the position [math]\displaystyle{ \vec{r} }[/math], so instead of getting the Coulomb field [math]\displaystyle{ \mathbf{E}=\frac{kq_e}{|\vec{r}|^2}\hat{r} }[/math] we get the electric field [math]\displaystyle{ \mathbf{E} = -\frac{q}{4\pi\epsilon_0} \left[ \frac{\mathbf{e}_{r'}}{r'^2} + \frac{r'}{c} \frac{d}{dt} \left(\frac{\mathbf{e}_{r'}}{r'^2}\right) + \frac{1}{c^2} \frac{d^2}{dt^2} \mathbf{e}_{r'} \right] }[/math], where [math]\displaystyle{ \vec{r}' }[/math] is the direction that the electron appears to be according to the observer, but is not actually. The magnetic field can then be written as [math]\displaystyle{ \mathbf{B} = -\frac{\mathbf{e}_{r'} \times \mathbf{E}}{c} }[/math]. Now if you notice the first two terms in the above E-field equation can be dropped at large [math]\displaystyle{ r }[/math] because they drop off with the distance squared. Calculating, we have that our electric field in a wave is [math]\displaystyle{ \mathbf{E} = -\frac{q}{4\pi\epsilon_0 c^2} \frac{d^2 \mathbf{e}_{r'}}{dt^2} }[/math]. The second derivative term is just the derivative of our velocity, so we get that [math]\displaystyle{ \mathbf{E} = \frac{q}{4\pi\|\vec{r}|epsilon_0 c^2}\sin(t) \vec{k} }[/math]. Using the B formula we get that since [math]\displaystyle{ r }[/math] is very large, the [math]\displaystyle{ e_r }[/math] vector and the [math]\displaystyle{ \mathbf{E} }[/math] vector are perpendicular, leading us to calculate B to be [math]\displaystyle{ \mathbf{B} = -\frac{q}{4\pi\epsilon_0|\vec{r}|c^3}\sin(t) \vec{i} }[/math]. So, from our investigations we conclude (by way of the Lorentz formula) that an oscillating point charge will cause a similar oscillation in the same direction over large distances.

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