# Equipotential Surfaces

## Equipotential Surfaces

The electric field can be characterised by using the vector quantity, electric intensity and also by using the scalar quantity, the potential. By using electric potential and potential energy, instead of electric intensity and electric force, many difficulties arising due to vector nature of electric field can be avoided. The flow of charge from one charged body to another, depends upon the quantity called electric potential. Electrical potential determines the direction of flow of charge from one point to another.

What is Equipotential Surface?

Any surface on which the electric potential is the same everywhere, is called an equipotential surface. It is the locus of points having the same potential due to a given charge distribution.

It is easier to locate equipotential lines that to measure electric fields directly. Through every point in an electric field, equipotential surface may be constructed.

Characteristics of Equipotential Surfaces:

1. Work done to move a test charge along an equipotential surface is zero, since any two points in it are at the same potential. i.e., potential difference between them is zero.

2. The electric intensity E is always perpendicular to the equipotential surfaces. The potential difference between any two points on an equipotential surface dV = 0

$$-\,\overrightarrow{E}.\overrightarrow{dl}\,\,=\,\,0$$,

– E dl cos θ = 0

E ≠ 0 And $$\overrightarrow{dl}\,\,\ne \,\,0$$,

So, cosθ = 0 or θ = 90°. Hence $$\overrightarrow{E}$$ is normal to the equipotential surface.

3. The equipotential surfaces help us to find the direction of the electric field.

4. The spacing between the equipotential surfaces will be lesser if the field is stronger and vice versa. Thus, we can distinguish stronger fields from weaker fields, using equipotential surfaces.

5. No two equipotential surface will intersect each other. If they intersect, then at the point of intersection there will be two values for the same potential, which impossible.

For an isolated point charge, the equipotential are concentric spheres. The field lines are perpendicular to the equipotential surfaces.

How to find the Potential Energy of an electron and a proton when separated at certain distance?

Problem: An electron and a proton are separated by a distance of 1.6 x 10⁻¹⁵ m. Calculate their potential energy?

Solution: When two charges q₁ and q₂ are separated by a distance of r, their mutual potential energy is given by:

$$U\,\,=\,\,-\,\frac{1}{4\pi {{\varepsilon }_{0}}}\times \frac{{{q}_{1}}{{q}_{2}}}{r}$$,

Where,

q₁ = + 1.6 x 10⁻¹⁹ C

q₂ = + 1.6 x 10⁻¹⁹ C

$$U\,=\,-\,\frac{(9\times {{10}^{9}})\times (-1.6\times {{10}^{-19}})\times (1.6\times {{10}^{-19}})}{1.6\times {{10}^{-15}}}\,=\,-\,1.44\times {{10}^{-13}}J$$,

Potential Energy (U) = – 1.44 x 10⁻¹³ J.