Equipotential Surfaces

An equipotential surface is a surface with a constant value of potential at all points on the surface. 

For a single charge q, the potential is given by, V = q/4πε0                

Fig. For a single charge q (a) equipotential surfaces are spherical surfaces centred at the charge, and (b) electric field lines are radial, starting from the charge if q > 0.

This shows that V is a constant if r is constant. Thus, equipotential surfaces of a single point charge are concentric spherical surfaces centred at the charge.  

Now the electric field lines for a single charge q are radial lines starting from or ending at the charge, depending on whether q is positive or negative.

Clearly, the electric field at every point is normal to the equipotential surface passing through that point. This is true in general: for any charge configuration, equipotential surface through a point is normal to the electric field at that point. 

Note:

If the charge is to be moved between any two points on an equipotential surface through any path, the work done is zero. hence electric field must be normal to the equipotential surface at every point. 

Equipotential surfaces offer an alternative visual picture in addition to the picture of electric field lines around a charge configuration.

Fig. Equipotential surfaces for a uniform electric field.

For a uniform electric field E, say, along the x -axis, the equipotential

surfaces are planes normal to the x -axis, i.e., planes parallel to the y-z plane (Fig.). 

Equipotential surfaces for (a) a dipole and (b) two identical positive charges are shown in Fig.

Fig. Some equipotential surfaces for (a) a dipole,

(b) two identical positive charges.