Equipotential surface and its properties

Electric potential at any point of the electric field tells us about its strength

It is a scalar quantity and can have the same value at different points in the electric field.

Electric potential at any point is the amount of work done in bringing a unit positive charge from infinity to that point.

By the definition, the potential at a point at distance from the charge is

For a point charge , the potential at a point will be given by

If we take other points in the space around at the same distance , the value of potential will be same as well

The set of these points equidistant from the point charge will constitute a surface where at each point the potential will be same.

This surface, where the value of potential is same at all points residing on it, is called the equipotential surface.

Thus, we can draw many equipotential surfaces around the charge , each at a particular distance

Let's discuss more about this equipotential surfaces.

Bringing a unit charge from one point to another in the electric field requires some work which is equal to the potential difference between the points

Thus, if we bring a charge from a point A to B we require to do some work.

But if we move the same charge on an equipotential surface, we don't have to do any work. As at the surface, the potential at each point is same.

As the potential difference is the amount of work that is required to move the object, the work done will be zero.

So, work done in displacing a charge on the equipotential surface will always be zero.

Let's understand it in terms of electric field lines

Electric field lines and equipotential surfaces

The electric field lines due to a positive point charge look like this

And let us consider an equipotential surface due to point charge

Considering field lines due to the point charge at the centre,

we can see field lines are emerging radially from the centre and will always be normal to the surface of the sphere

If direction of electric field is not normal to the equipotential surface, it would have components along the equipotential surface

For which, we need to do work for moving the charge on the equipotential surface, which is not true as we have discussed

So, at the equipotential surfaces, the electric field lines will always be perpendicular to it

Revision

An equipotential surface is a surface with a constant value of potential at all points that lie on it.

Work done in displacing a charge on an equipotential surface is always zero.

Electric field lines are always normal to the equipotential surfaces.

The End