# Equipotential Surface

A surface having the same potential at every point is referred to as an equipotential surface. There is no work done in order to move a charge from point A to B on equipotential surfaces.

The points present in an electric field having similar electric potential are called equipotential points

When similar potential points are connected by a curve or a line, they are referred to as an equipotential line, and when points lie on a surface, they are called an equipotential surface.

If these points are distributed throughout either volume or space, it is called an equipotential volume.

Index

## Work Done on Equipotential Surface

The work done in moving a charge between the two points in an equipotential surface is zero since the potential is the same at all points on that surface.

If a point charge is being moved from the point $$A$$ having potential $$V_A$$ to $$B$$ having potential $$V_B$$, the work done in moving the charge can be given by,

$$W = q_o(V_A – V_B)$$

As the potential of  $$V_A = V_B$$, the total work done, $$W = 0$$

## Properties of Equipotential Surface

The properties of the equipotential surfaces are:

1. Two equipotential surfaces never intersect.
2. The electric field and equipotential surfaces are perpendicular to each other.
3. For a uniform electric field, the equipotential surfaces are planes that are normal to the x-axis.
4. The equipotential surface directions from high potential to low potential.
5. For a point charge, the equipotential surfaces will be in the form of concentric spherical shells.
6. In a uniform electric field, the planes normal to the direction of the field are equipotential surfaces.
7. The potential inside a hollow charged spherical conductor is a constant. This is treated as equipotential volume. (No work is done in moving a charge from the center to the surface.)
8. For an isolated point charge, the equipotential surfaces are in the form of a sphere. I.e.,  the concentric spheres around the point charge will contain different equipotential surfaces.
9. The space between the equipotential surfaces allows us to identify strong and weak field regions.
$$E = \frac{-d_v}{d_r} \Rightarrow E \propto \frac{1}{dr}$$

## Example Problems

Question 1. A charged particle ($$q = 2mC$$) moves a distance of 2 m along an equipotential spherical shell of 7 V, what is the work done by the field during this motion.

Solution. The work done is given by $$W = -q \Delta V$$

Since, $$\Delta V = 0$$
Therefore, $$W = 0$$

Question 2. A positive particle of charge 1.0 C accelerates in a uniform electric field of 10 V/m. The particle started from rest on an equipotential plane of 5 V. After t = 0.00002 seconds, the particle is on an equipotential plane of V = 1 volts. Determine the distance traveled by the particle.

Solution. We know that, $$W = -q \Delta V$$

$$W = (-1.0 C) (1V – 5V) = 4 J$$

Work done in moving a charge in an electric field: $$W = qEd$$

$$4 = (1.0) (10)d$$

$$d = 0.4m$$

Thus, the distance traveled is 0.4 m.

## FAQs

What is an equipotential surface?

A surface having the same potential at every point is referred to as an equipotential surface.

What is Work done on an Equipotential surface?

The work done in an equipotential surface is Zero.

What is Equipotential Volume?

Equipotential points that are distributed throughout either volume or space, it is called equipotential volume.

Equipotential surface for a point?

A point charge will form an equipotential spherical shells.

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