Maxwell's differential equations allow the electromagnetic fields to be represented by a vector potential $\mathbf A$ and a scalar potential $\phi$. The potential representation builds two of Maxwell's equations into the definitions themselves.

Magnetic field from a vector potential

Gauss's law for magnetism states

$$\nabla\cdot\mathbf B=0.$$

On a suitable region with no relevant topological obstruction, a divergence-free magnetic field can be represented as

$$\boxed{\mathbf B=\nabla\times\mathbf A}.$$

This automatically satisfies the magnetic divergence equation because the divergence of a curl vanishes.

Electric field from the potentials

Faraday's law is

$$\nabla\times\mathbf E =-\frac{\partial\mathbf B}{\partial t}.$$

Substituting $\mathbf B=\nabla\times\mathbf A$ gives

$$\nabla\times\mathbf E =-\nabla\times\frac{\partial\mathbf A}{\partial t},$$

so

$$\nabla\times\left( \mathbf E+rac{\partial\mathbf A}{\partial t} \right)=0.$$

The curl-free-field criterion then guarantees a local scalar potential. Choosing the conventional sign,

$$\mathbf E+rac{\partial\mathbf A}{\partial t}=-\nabla\phi,$$

and therefore

$$\boxed{ \mathbf E =-\nabla\phi

\frac{\partial\mathbf A}{\partial t}}. $$

In electrostatics, the fields are time-independent and the second term vanishes, recovering

$$\mathbf E=-\nabla\phi.$$

For time-dependent electromagnetic fields, a scalar potential alone is generally insufficient.

Potentials contain redundant information

The fields $\mathbf E$ and $\mathbf B$ do not determine a unique pair $(\phi,\mathbf A)$. Different potentials can produce exactly the same physical fields. This gauge freedom is not a defect: it allows a convenient representation to be chosen without changing observable electric or magnetic fields.

With an appropriate gauge choice, the remaining Maxwell equations become equations for $\phi$ and $\mathbf A$ that resemble driven wave equations. Their solutions make the finite propagation time of electromagnetic influences explicit.

The potentials therefore provide more than an alternative notation. They turn the homogeneous Maxwell equations into structural identities and provide the natural variables for electromagnetic radiation, relativistic formulations, and quantum interactions.