Define an equipotential surface. Draw equipotential surfaces:

(i) in the case of a single point charge and

(ii) in a constant electric field in Z-direction. Why the equipotential surfaces about a single charge are not equidistant?

(iii) Can electric field exist tangential to an equipotential surface? Give reason.

An equipotential surface is a surface on which potential is same everywhere, or it is constant; there is no change in potential at different points that lie on the same equipotential surface, difference or change in potential is

ΔV = 0 V

Work done in moving a charge from one place to another on an equipotential surface is zero, Electric Field is always perpendicular to an equipotential surface, so a same system of charge there can be many equipotential surfaces but two equipotential surfaces never intersect each other, and they should be drawn such that electric field is perpendicular to them.

(i) we know in case of point charge electric field is spherically symmetric in nature , i.e. at same distance from point charge at different points, the electric field has the same magnitude and is always directed radially outward (positive charge) or radially inward (negative Charge). So equipotential surface would be concentric spheres with the center as the point charge, because all the points on the sphere are equidistant from point charge and normal to the surface is in the same direction as of electric field, or we can say electric field at all point is normal to surface

as shown in figure

(ii) Now in case of constant electric in Z-direction, electric field lines will be parallel to the Z axis, so any parallel to X-Y plane or perpendicular to Z axis will be an equipotential surface as electric field lines will always be perpendicular to the plane

As Shown in Figure

Now the equipotential surfaces about a single charge are not equidistant, this means suppose we move from one equipotential surface having constant potential of 9 volts on its surface to other equipotential surface with constant potential 6 volts if we covered a covered a distance r then to reach an equipotential surface having constant potential of 3 volts ,we have to cover a greater distance i.e. spacing between surfaces or distance is not same because for an point charge electric potential is given as

V = Kq/r

Where V is potential at a distance of r from a charge of magnitude q and K is the Coulumb’s constant

So as we can see as distance increase potential decrease non linearly as the potential is inversely proportional to the distance from the center

(iii) No, Electric field cannot be tangential to the equipotential surface as

If the fields are not normal to the equipotential surface, it would

Moreover, have a non-zero component along the surface. To move a unit test charge against the direction of the component of the field, work would have to be done. As work done is given by

W = FScos𝜽

Where W is the work done, S is displacement F is the force and 𝜽 is the angle between force and displacement

For work done to be zero angles between force and displacement must be ninety degrees as cos𝜽 = 0 when 𝜽 = 90⁰

Force on a charged particle is in the direction of electric filed so if electric filed is a tangential force is tangential to surface on which particle is displaced, hence angle will be 𝜽 = 0⁰

So cos𝜽 = 0

i.e. work done will not be Zero but a finite value

However, this is in contradiction to the definition of an equipotential

Surface there is no potential difference between any two points on the surface, and hence no work is required to move a charge on the surface. The electric field must, therefore, be normal to the equipotential surface at every point and not tangential.