# Introduction to Equipotential Surface

When you find a constant potential upon any part of the surface is known as an equipotential surface. We can also define it as the result of the potential difference between any two points is zero. A surface stands for the locus of all points. Each point delivers the same potential; then, it is called equipotential.

Have you ever wondered what is equipotential surface is? In this type of condition, you should put no effort to move a charge from one place to another. An equipotential surface has the benefit that each point has the same potential level.

### Define Equipotential Surface

In other terms, an equipotential surface is a surface that exists with the same electrical potential at each point. If any point lies at the same distance from the other, then the sum of all points will create a distributed space or a volume. Scientists have termed it as equipotential volume.

Do you know the work done on an equipotential surface?

Well, now it’s time to know the truth. Let’s consider that VA and VB both are two point charges in an equipotential surface. We need to calculate the work done in the moving charge. The relation is given below:

W = q0(VA –VB)

However, on an equipotential surface, you will find the difference between two significant points is zero.

That means VA –VB = 0

So, work done = 0

### Properties of Equipotential Surface

Here is the list that can help you to know the equipotential surface properties:

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

2. The chance of the intersection of two equipotential surfaces is impossible.

3. Equipotential surfaces look like concentric spherical shells for a point charge.

4. When the Equipotential surfaces are under a uniform electric field, they act as normal towards the x-axis.

5. The potential of a hollow charged spherical conductor is constant. This is also named equipotential volume. There will be no work done to move a charge from its centre.

6. The equipotential surface acts as a sphere when the point charge is isolated. This says that concentric spheres that revolve around the point charge possess different equipotential surfaces.

7. The equipotential surface follows the direction from higher potential to lower potential.

8. When any plane is available normal to the uniform electric field acts as an equipotential surface.

9. Gaps that are available between equipotential surfaces help us to determine whether the regions are stronger or weaker.

Mathematically, E= −dV / dr ⇒ E ∝ 1/dr

### Some Problems on Equipotential Surface

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 ×108 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 = $\frac{F}{m}$ = $\frac{1.6 \times 10^{-13}}{9.6 \times 10^{-31}}$ = 1.8 * 1017 m/sec2

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 = $\frac{v}{a}$ = $\frac{0.1 \times c}{a}$ = $\frac{0.1 \times(3 \times 10^{8})}{1.8 \times 10^{17}}$

### Some Basic Ideas about Equipotential Surface

We can represent the equipotential process with ease. Electric potentials or Voltage is the form of electric fields. Both of the terms, such as Voltage and equipotential surface, are interrelated. They are very significant terms that can help you to learn.

You don’t have to put effort or supply any power to move a charge upon equipotential surfaces, as here you will find the change in Voltage is zero.

W = −ΔPE = −q ΔV = 0

So, all the efforts (work) will be just a waste of time, isn’t it! When You notice that the field in motion is perpendicular with force, the work is also zero.

When the force F acts in the same direction as E (electric field), the motion is always lying perpendicular to E.

The relation is, W = Fd cos θ = qEd cos θ = 0.