Equipotential surfaces are surfaces over which the electric potential is constant. This means that if you were to move a test charge along an equipotential surface, no work would be done by the electric field.

The electric field is always perpendicular to equipotential surfaces. This is because the electric field points in the direction of the force that would be exerted on a positive test charge. If the electric field were not perpendicular to the equipotential surface, then the test charge would move along the equipotential surface, and work would be done by the electric field.

The expression for the electric field as a negative of the potential gradient is:

E = -∇V

where:

* E is the electric field (in volts per meter)
* ∇ is the gradient operator
* V is the electric potential (in volts)

The gradient operator is a mathematical operator that takes a scalar field and returns a vector field that points in the direction of the greatest increase of the scalar field. In this case, the scalar field is the electric potential.

The negative sign in the expression indicates that the electric field points in the direction of decreasing potential.

Equipotential surfaces are a useful way to visualize the electric field. They can be used to understand how the electric field is distributed around a charged object.

Here are some examples of equipotential surfaces:

* The surface of a charged sphere is an equipotential surface.
* The space between the plates of a capacitor is an equipotential surface.
* The electric field lines around a point charge form concentric circles, with the equipotential surfaces being perpendicular to the electric field lines.