Electromagnetism (Maxwell's equations)

What is it?

Formulas

$$\nabla\cdot E = \frac{\rho}{\varepsilon_0}, \quad \nabla\cdot B = 0, \quad \nabla\times E = -\frac{\partial B}{\partial t}, \quad \nabla\times B = \mu_0 J + \mu_0\varepsilon_0\frac{\partial E}{\partial t}$$

The mathematics behind it

• Vector fields★★★★★fundamental

  Electric and magnetic fields $E$ and $B$ are vector fields.

• Divergence★★★★★fundamental

  Two of Maxwell's equations are divergence equations: $\nabla\cdot E = \rho/\varepsilon_0$, $\nabla\cdot B = 0$.

• Curl★★★★★fundamental

  Faraday's and Ampère's laws are curl equations; induction in motors and antennas follows from them.

• Divergence theorem (Gauss)★★★★★fundamental

  Gauss's law in integral form: the flux of $E$ through a closed surface is the enclosed charge over $\varepsilon_0$.

• Stokes' theorem★★★★★fundamental

  A changing magnetic flux through a loop induces a voltage around it — generators and transformers.