## An Introduction to Equipotential Surfaces

The surface which is the center of all points and has the same potential is called the equipotential surface. On the equipotential surface, to move a charge from one point to another no work is required.

Equipotential Points: The same electric potential points on the electric field are called equipotential points. The line or curve connecting the points is known as an equipotential line. The surface on which the point lies is called the equipotential surface. The volume in which the points are filled is known as an equipotential volume.

### Work Done in an Equipotential Surface

In an equipotential surface, the work done to move a charge between any two points will always be zero. In an equipotential surface, if a point charge is said to move from point V_{A} to V_{B}, then the work done in moving the charge is given by,

W = q_{0} (V_{A} –V_{B})

As V_{A} – V_{B }is said to be zero and the total work done, W = 0.

### Properties of Equipotential Surface

The electric field and equipotential surface are always perpendicular to each other.

Two equipotential surfaces will not be intersected.

The equipotential surfaces are said to be concentric spherical shells for a point charge.

The equipotential surfaces are planes which are normal to the x-axis for an uniform electric field.

The direction of an equipotential surface will always range from high potential to low potential.

The potential is constant inside a charged hollow spherical conductor. This is called an equipotential volume. In general, no work is necessary to move a charge from the center to the surface.

The equipotential surface is said to be a sphere for an isolated point charge.

Any plane which acts normal to the field direction is referred to as an equipotential surface in a uniform electric field.

Strong and weak fields can be identified using the space between equipotential surfaces i.e. E= −dV/dr ⇒ E ∝ 1/dr

### Problems

Q1. A Charged particle possessing a 1.4 mC charge travels a distance of 0.4m on an Equipotential Surface. The Voltage that the Equipotential Surface carries is 10 V. Find out the Work Done by the Field during the motion of the charged Particle.

Ans: Work done over the electric field in an equipotential surface has a Mathematical expression:

W = -q ΔV

As we know, work done will be zero here because, in this case, ΔV = 0.

Therefore, W = 0

Q2. If an Electric Charge having a certain mass and charge is free from equilibrium to motion. It Interacts with the uniform electric field of 160 N/C. Calculate the amount of time taken by the electron in achieving the speed of 0.1 c. Given: c = the Velocity of Light (c = 3 ×10^{8} meter/sec); Mass of the Electron m = 9.1 × 10^{–31} kg, e = 1.6 × 10^{–19} Coulomb.

Ans: The amount of force applied over the electron,

F = E * e

= 106 * (1.6 × 10^{–19}) N

= 1.6 * 10^{-13} N

We can find the acceleration of the electron by the following method:

a = F/m

= \[\frac{1.6×10^−{13}}{9.6×10^{−31}}\]

= 1.8 * 10^{17} m/sec^{2}

As mentioned in the question, the initial velocity is zero

Consider the time taken by the electron to achieve a speed of 0.1 c.

So, we can insert the formula

v = u + at

or v = at

or, t = v/a

= \[ 0.1× \frac{c}{a}\]

= \[ 0.1× \frac{(3×10^8)}{1.8×10^{17}}\]

## FAQs on Introduction to Equipotential Surface

**1. What is an equipotential surface?**

The surface which is the center of all points and has the same potential is called the equipotential surface. To move a charge from one point to another no work is required on the equipotential surface.

Equipotential Points: The same electric potential points on the electric field are called equipotential points. The line or curve connecting the points is known as an equipotential line. The surface on which the point lies is called the equipotential surface. The volume in which the points are filled is known as an equipotential volume.

**2. List some of the properties of an equipotential surface?**

The electric field and equipotential surface are always perpendicular to each other.

Two equipotential surfaces will not be intersected.

The equipotential surfaces are concentric spherical shells for a point charge.

The equipotential surfaces are planes normal to the x-axis for a uniform electric field.

The direction of the equipotential surface ranges from high potential to low potential.

Inside a charged hollow spherical conductor the potential is said to be constant. This is called an equipotential volume. To move a charge from the center to the surface, no work is required.

The equipotential surface is said to be a sphere for an isolated point charge.

Any plane normal to the field direction is an equipotential surface in a uniform electric field.

**3. Does the electric field exist tangentially to equipotential surfaces? **

No, As we know that Potential difference along the Electric field = E*d, where E is the magnitude of the electric field and d is the distance and direction of movement of the electric field from a fixed point. Now assume if there is an electric field along the surface of Equipotential Surface then as per the equation there must be some potential difference. Therefore, the surface will not be an equipotential surface. That is the reason for no field along the tangent to the surface of the equipotential surface.

**4. Prove that a closed equipotential surface which has no charge must enclose an equipotential volume?**

For instance, let us consider a contradicting statement that the potential is not the same inside, the closed equipotential surface.

Let the potential just inside the surface be different from that on the surface having a potential gradient (dV/dr).

Consequently, an electric field comes into existence, which is given by, E = -dV/dr

Similarly, the field lines get pointed inwards or outwards from the surface.

As the surface is equipotential, these lines cannot be formed on the surface.

It is possible only if the opposite side of the field lines originate from the charges inside.

This contradicts the original assumption.

Thus, the total volume must be equipotential.

**5. From the following which is wrong in equipotential surface on an uniform electric field?**

**(a) Equipotential surface is flat**

**(b) Work done is zero**

**(c) Equipotential surface is spherical**

**(d) Electric lines are perpendicular to the equipotential surface**

(c) Equipotential surface is spherical

**Reason:** The equipotential surfaces are planes that are normal to the x−axis for a uniform electric field along the x- axis. As, on the equipotential surface, the potential at all points is constant, this implies, work done to move a charge from one point to another is zero.