Question 7: Describe the concept of equipotential surfaces and derive an expression for electric field as negative of potential gradient.

Equipotential surfaces/lines

An equipotential surface (or line) connects all those points in space where the potential due to an electric field is same. For a point charge
Since (q0/4πε0) is constant for a particular charge, therefore, electric potential depends upon the distance from the point charge. So at all equidistant points the potential would have same value. Since electric field due to a point charge extends in all directions (three dimensions) therefore, the equipotential surfaces are in the form of concentric spheres with the point charge at the center. However, if we deal with a two-dimensional case, concentric circular lines around the point charge make equipotential lines. Equipotential surfaces/lines have some important characteristics:
1. No two surfaces cross each other.
2. No work is done in moving a charge from one point to another on the equipotential surfaces or lines.
3. Electric potential is a scalar quantity. However, potential may be positive or negative depending upon on the sign of charge producing it.
4. If there are two or more charges, the electric potential at a particular point is the scalar sum of potentials due to individual charges.
5. If the electric field is stronger, the lines on the equipotential surface would be close together. So the electric potential energy which is inversely proportional to r is changing by a large amount in small distances and there must be a large force acting.
Expression for electric field as negative of potential gradient