Maxwell's equations are really important in the study of electromagnetism. They explain how electric fields, magnetic fields, and charges work together. These four equations show us how electric charges create electric fields, how electric currents make magnetic fields, and how changing electric and magnetic fields can travel as electromagnetic waves.

The Four Maxwell's Equations

1. Gauss's Law for Electricity: This equation says that electric fields spread out from positive charges and come together at negative charges.

2. Gauss's Law for Magnetism: This tells us that there are no single magnetic charges, called monopoles. Instead, magnetic field lines always loop back around or stretch out endlessly.

3. Faraday's Law of Induction: This means that when a magnetic field changes over time, it creates an electric field.

4. Ampere-Maxwell Law: This shows us how electric currents and changing electric fields produce magnetic fields.

How Waves Move

Now, let's see how electromagnetic waves move. To do this, we can use Maxwell's equations. Let’s think about a space without charges or currents.

Starting with Faraday's Law and adding it to the Ampere-Maxwell Law, we can find out how the electric field behaves:

1. We take a special math operation called the curl of both sides of Faraday's Law.

2. Using some math tricks, we get an equation for the electric field's wave: This ends up being a wave equation. It tells us how the electric field changes as it moves.

We can also find a similar equation for the magnetic field.

Speed of Electromagnetic Waves

Both equations show that electromagnetic waves travel at a speed known as $c$. This speed is calculated as: $c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$

This is actually the speed of light in empty space, which is about (300,000,000) meters per second!

Example: Plane Waves

One way these waves can be represented is with plane waves, which can be written like this: $\mathbf{E}(z, t) = E_0 \cos(kz - \omega t)$ $\mathbf{B}(z, t) = B_0 \cos(kz - \omega t)$

Here, (E_0) and (B_0) are the strengths of the electric and magnetic fields, (k) is related to how many waves fit in a certain space, and (\omega) tells us how fast the wave cycles.

In short, Maxwell's equations connect electric and magnetic fields and show us how changes in these fields can cause electromagnetic waves to move. This is one of the coolest parts of physics!