Electromagnetic Wave

• An EM wave can be created from an accelerating charge?

Definition §

An electromagnetic wave is a transverse wave consisting of an Electric Field and a Magnetic Field such that:

• $\vec{B}$ and $\vec{E}$ are both perpendicular to the direction of propogation, given by ${\vec{v}}_{EM}$. I.e.

$${\vec{v}}_{EM}=\vec{E}\times \vec{B}$$

• The wave travels at the speed of light at all times:

$$\Vert {\vec{v}}_{EM}\Vert =c$$

• The magnitude of the electric field is always the product of the speed of light and the magnitude of the magnetic field:

$$\Vert \vec{E}\Vert =c\Vert \vec{B}\Vert$$

Correlation of Electromagnetism in Free space §

Free space is defined to be a region without any matter, i.e. a vacuum. In free space, we have no particles to act as charge carriers, and hence we have an absence of charges.

Now, look at how Faraday’s Law and Ampère-Maxwell Law correlate with each other:

$$\oint {\vec{E}}_{ind}\cdot d\vec{l}=-\frac{\text{d}}{\text{d}t}\left(\oint \vec{B}\cdot \text{d}\vec{A}\right)$$

$$\oint \vec{B}\cdot d\vec{l}={\mu }_0\left(I+{ϵ}_0\frac{\text{d}}{\text{d}t}(\oint \vec{E}\cdot d\vec{A})\right)$$

In a vacuum, $q=0$ which implies $I=\frac{\text{d}q}{\text{d}t}=0$:z

$$\oint \vec{B}\cdot d\vec{l}={\mu }_0(\cancel{I}+{ϵ}_0\oint \vec{E}\cdot d\vec{A})={\mu }_0{ϵ}_0\frac{\text{d}}{\text{d}t}(\oint \vec{E}\cdot d\vec{A})$$

And it looks like we get a very nice correlation:

$$\oint {\vec{E}}_{ind}\cdot d\vec{l}=-\frac{\text{d}{\Phi }_B}{\text{d}t}$$

$$\oint \vec{B}\cdot d\vec{l}={\mu }_0{ϵ}_0\frac{\text{d}{\Phi }_E}{\text{d}t}$$

It shows that a changing Magnetic Field induces an electric field, and the same applies for a changing Electric Field.

Also know that ${\mu }_0{ϵ}_0=\frac{1}{c^2}$ ($c$ being the speed of light)#tosee why

Frames of reference §

We can use frames of reference as a means to observe the relationship between electric and magnetic fields.

Take an isolated system consisting of a positively-charged particle at uniform velocity, moving in a constant uniform magnetic field that’s perpendicular to its velocity. Because we start by assuming the velocity is constant, we have an inertial frame of reference.

Recall that forces must be equal in inertial frames of reference.

In Frame 1, there is a Magnetic Force acting on the particle given by:

In Frame 2, however, the particle is stationary. Do we even have a force?

We know that because these are both inertial frames of reference, we must have a force that is the same magnitude and direction as that in Frame 2.

For now, we’ll assume that this is caused by an induced electric field. We’ll prove it in the next section, but for now, just trust me.

An Induced Electric Field is created. From Ampere-Maxwell Law we know that:

And the direction of this electric field is such that it’s magnetic field opposes our original magnetic field. Now, we have an Electric Force acting on the particle, given by:

We need both forces to be equal (as forces in inertial frames are equal):

$$F_E=F_B\to q\vec{E}=\vec{B}q\times \vec{v}$$

$$\vec{E}=\vec{B}\times \vec{v}$$

Proof of Direction §