Equipotential Surface

Equipotential Surface

A real or imaginary surface in an electric field that has the same potential at every point is called an equipotential surface.

Equipotential surfaces can be of any shape.

For example for a point charge of having charge q the potential at a distance, r is given as $V=\frac{kq}{r}$

So For V=constant, we get r=constant means for a point charge having charge q, the equipotential surfaces are the concentric spherical surfaces having a charge q at their centre as shown in the below figure.

All points on the spherical surface of radius r centred on q have the same V.

Properties of the equipotential surface-

• The potential difference between any two points on the Equipotential surfaces is zero.
• No work is done by the electric force to move the charge from one point to another point on an equipotential surface.
• Equipotential surfaces can never cross each other, otherwise potential at a point will have two values which is not possible.
• An equipotential surface is always perpendicular to electric lines of force.

For example, An equipotential surface for a uniform electric field is shown below.

From the figure, it is clear that the Direction of the electric field is perpendicular to the equipotential surface.

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Solved Examples Based On Equipotential Surface

Example 1: Equipotential surfaces associated with an electric field which is increasing in magnitude along the x-direction are

1)Planes parallel to yz-plane

2)Planes parallel to xy-plane

3)Planes parallel to xz-plane

4)Coaxial cylinders of increasing radii around the x-axis

Solution:

As we have learnt,

Equipotential Surface -

All Points have the same Potential.

Example 2: The points resembling equal potentials are

1)P and Q

2)S and Q

3) S and R

4)P and R

Solution: An equipotential surface is always perpendicular to electric lines of force.

From the above figure, it is clear that the line SR is perpendicular to electric lines of force. i.e The line SR represents the equipotential line. Hence Potential along the line SR at all points will remain constant.

So the points resembling equal potentials are S and R.

Example 3: The angle between the equipotential surface and lines of force is

1)Zero

2)$180^\circ$

3) $90^\circ$

4)$45^\circ$

Solution:

As we have learnt,

Equipotential Surface -

The Direction of the electric field is Perpendicular to the equipotential surface.

Lines of force are perpendicular to the equipotential surface. Hence angle = 90^o

Example 4: Equipotential surfaces associated with an electric field which is increasing in magnitude along the x-direction are

1) Planes parallel to yz-plane

2)Planes parallel to xy-plane

3)Planes parallel to xz-plane

4)Coaxial cylinders of increasing radii around the x-axis

Solution:

As we have learnt,

Equipotential Surface -

Never cross each other.

Because the angle between the electric field and the equipotential surface is 90^o

Example 5: A hollow metallic sphere of radius R is given a charge Q. Then the potential at the centre is

1)Zero

2) $\frac{1}{4\pi\epsilon_0}\frac{Q}{R}$

3)$\frac{1}{4\pi\epsilon_0}\frac{2Q}{R}$

4)$\frac{1}{4\pi\epsilon_0}\frac{Q}{2R}$

Solution:

As we have learnt,

Equipotential Surface -

Never cross each other.

Potential V anywhere inside the hollow sphere, including the centre is

$V = \frac{1}{4\pi\epsilon_0}\frac{Q}{R}$

Summary:

Any imaginary surface in an electric field for which every point is at the same electric potential is called an equipotential surface. They are always at right angles to the electric field lines. Due to different charge distributions, these surfaces take various shapes. For instance, they are spherical around a point charge and planar between parallel plates. Equipotential surfaces are a way to simplify the investigation of electric fields because there is no need for any work to be done for transporting a charge on those surfaces.