# Equipotential Surface

## What is an Equipotential Surface?

The locus of all point’s surfaces, which exists at a similar potential, is called the equipotential surface. There is no work essential to move a charge from one point to another on the equipotential surface. In simple words, any surface having the same electric potential at every point is referred to as an equipotential surface.

If the points present in an electric field are all at similar electric potential, then they are called the equipotential points. If these points are connected by a curve or a line, it is referred to as an equipotential line. When such points lie on a surface, it is called an equipotential surface. Furthermore, if these points are distributed throughout either a volume or space, it is called an equipotential volume.

### Work Done in Equipotential Surface

The work done happens while moving a charge between the two points in an equipotential surface is zero. If a point charge is moved from the point VA to VB in an equipotential surface, the work done in moving the charge can be given by,

W = q0(VA –VB)

Because VA – VB value is equal to zero, the total work done, W = 0.

### Properties of Equipotential Surface

The main properties of the equipotential surface are listed below.

• Two equipotential surfaces can never intersect

• Always, the electric field is perpendicular to an equipotential surface.

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

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

• The potential is constant inside a hollow charged spherical conductor. This can be treated as equipotential volume. No work is needed to move a charge from the centre to the surface.

• The equipotential surface direction is from high potential to low potential.

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

• The equipotential surface is a sphere for an isolated point charge. It means that the concentric spheres around the point charge contain different equipotential surfaces

• The spacing between the equipotential surfaces enables us to identify strong and weak field regions. It means, E= −dV/dr ⇒ E ∝ 1/dr

### Let’s Look Into an Example Problem on the Equipotential Surface.

If an electron having mass ‘m’ and charge ‘e’ is released from rest in a uniform electric field of 106 newton/coulomb, then compute its acceleration. Find the time taken by the electron in attaining a speed of 0.1 c, where ‘c’ is given as velocity of light. (m = 9.1 × 10–31 kg, e = 1.6 × 10–19 coulomb and c = 3 ×108 metre/sec).

F = Ee = 106 (1.6 × 10–19)

= 1.6 × 10–13 newton

Similarly, the acceleration of the electron is given as,

a = F/M = (1.6 x 10-13)/(9.110-31)

= 1.8 x 1017 m/sec2

The initial velocity is zero. Let ‘t’ be the time taken by the electron in attaining a final speed of 0.1 ‘c.’

Now, v = at or v = u + at

Therefore, t = v/a= (0.1 x c)/a = $\frac{0.1 \times (3 \times 10^{8}) }{1.8 \times 10^{17}}$

= 1.7 10-10sec

### Electric Equipotential Surface

In the case of n-dimensional space (for instance, in 2-dimensional space), if the electric field lines are present, then the equipotential surface lies perpendicular to this plane, and therefore this imaginary surface lies along the Z-axis if the electric field is set in an X-Y plane.

Hence, the equipotential surface is always perpendicular to the electric field lines. This surface is maintained at an equal potential as there is no change expected along this surface.

When we consider an equipotential surface, then we should also consider what would be the work done if we move charge along this surface.

Now, work done in electrostatics is given by,

W = −q x ΔV = −ΔU

Here, q is the charge, and ΔV is the change in potential.

ΔU  is the electric potential energy gained by a charge when it is forced to move in external electric potential.

As work is done on the expense of electric potential energy, thus, W = −ΔU

As the name says, surfaces are equipotential, resultantly, there is no change in electric potential, and thus no energy is gained by the charge.

As energy is not gained, thus no work has been done in moving charge along the equipotential surface.

### Magnitude of the Electric Field on the Equipotential Surface

The electric field in the equipotential surface direction (in the direction, which is parallel to that surface) is zero. That is less or more the definition of an equipotential surface because the integral line of the electric field is the potential. So, for the potential to stay the same, the field must be zero.

1. Explain the Equipotential Surface in Detail.

We can represent the electric potentials or voltages pictorially, just as we drew the pictures to illustrate the electric fields. Since the two concepts are related, this is not surprising.

The above representation shows an isolated positive point charge and its electric field lines that radiate out from a positive charge, whereas, it terminates on negative charges. Here, we used blue arrows to represent the direction and magnitude of the electric field, and green lines to represent the places where the electric potential remains constant.

These are known as the equipotential surfaces in three dimensions or the equipotential lines in two dimensions. The term equipotential can also be used as a noun, referring to an equipotential surface or line. At any point, the potential for a point charge is the same on an imaginary sphere of radius r surrounding the charge. This is absolutely true because the potential for a point charge is given as below.

V=kq/r

Therefore, it has the same value at any point at a given distance ‘r’ from the charge. Due to the electric field lines point away from the charge radially, they remain perpendicular to the equipotential lines.

2. Can Two Equipotential Surfaces Intersect?

The two equipotential surfaces cannot intersect because it would contradict how an equipotential surface is defined.

Each specific equipotential surface is defined as the set of all points within a region of space that shares a common potential value. If two of these surfaces intersect, it would mean that the points of intersection have two different potential values, which in turn makes no sense in any situation (where an equipotential surface is useful at all).

If we have the distributions with two different charges, each with their own set of equipotential surfaces, and we bring them near to each other. The surfaces don’t intersect the shift form to reflect the new configuration charge.