The electromagnetic potentials are not unique. If a differentiable scalar function $\chi(\mathbf r,t)$ is chosen, the transformation $$\mathbf A' = \mathbf A+\nabla\chi,$$ $$\phi' = \phi-\frac{\partial\chi}{\partial t}$$ leaves the physical electric and magnetic fields unchanged.

For the magnetic field, $$\nabla\times\mathbf A'= \nabla\times\mathbf A+\nabla\times\nabla\chi =\mathbf B,$$ because the curl of a gradient is zero. For the electric field, $$-\nabla\phi'-\frac{\partial\mathbf A'}{\partial t} =-\nabla\phi-\frac{\partial\mathbf A}{\partial t},$$ because the two derivatives of $\chi$ cancel.

This redundancy is called gauge freedom. A gauge condition selects one representative from the many potential pairs describing the same fields.

The Coulomb gauge imposes $$\nabla\cdot\mathbf A=0$$ and is often convenient in problems where electrostatic structure is prominent. The Lorenz gauge imposes $$\nabla\cdot\mathbf A+\frac{1}{c^2}\frac{\partial\phi}{\partial t}=0,$$ which treats space and time in a form naturally compatible with special relativity and turns the potential equations into wave equations sourced by charge and current.

Choosing a gauge does not change experimental predictions. It changes the mathematical representation, much as choosing coordinates changes component values without changing the underlying geometry.

Gauge freedom becomes even more important in quantum and particle physics, where local gauge symmetry organizes the form of fundamental interactions. Classical electromagnetism provides the simplest concrete example of this broader principle.