Electromagnetism is formulated in terms of the electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$. Charges produce electric fields, currents produce magnetic fields, and time-varying fields generate each other. This mutual coupling is captured by Maxwell's equations.

In differential form, Maxwell's equations are $$\nabla\cdot\mathbf{E}=\rho/\varepsilon_0, \qquad \nabla\cdot\mathbf{B}=0,$$ $$\nabla\times\mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}, \qquad \nabla\times\mathbf{B}=\mu_0\mathbf{J}+\mu_0\varepsilon_0\frac{\partial \mathbf{E}}{\partial t}.$$ Together they imply the existence of electromagnetic waves.

The fields can be written in terms of potentials: $\mathbf{B}=\nabla\times\mathbf{A}$ and $\mathbf{E}=-\nabla \phi - \partial \mathbf{A}/\partial t$. Different choices of $\phi$ and $\mathbf{A}$ can represent the same physical fields, a redundancy known as gauge freedom.