Equipotential Surfaces

Electrostatic field lines help us visualize electric fields. Similarly, potential at various points in an electric field can be represented graphically by equipotential surfaces.

To help mountain climbers, contour lines are drawn on topographic points through points of some elevation. By analogy to contour lines on a topographic map, an equipotential surface is a three dimensional surface on which the electric potential 'V' is the same at every point. In a region where an electric field is present, equipotential surfaces can be constructed. If a test charge q_o is moved from point to point on such a surface, the electrical potential energy q_ov remains constant.

Properties of equipotential surfaces

1) No point can be at different potentials, so equipotential surface for different potentials can never touch or intersect.

2) Potential energy does not change as a test charge moves over an equipotential surface and hence the electric field can do no work on such a charge. Because of this reason must be perpendicular to the surface at every point so that the force q_o will always be perpendicular to the displacement of a charge moving on the surface. Hence, field lines and equipotential surfaces are always mutually perpendicular.

3) Equipotential surfaces are drawn so that there are equal potential differences between adjacent surfaces.

In regions where the magniture of is large, the equipotential surfaces are closer together because the field does a relatively large amount of work on a test charge in a relatively small displacement. In regions where the field is weaker, the equipotential surfaces are farther apart.

4) On a given equipotential surface, the potential V has the same value at every point but in general the electric field does not. For example, at the midpoint of the line joining a dipole, potential is zero but the electric field is not zero.

Relationship between Electric Field and Potential

 

 

The above figure shows the electric field of a positive point charge.

The electric field is directed away from the charge and potential is positive at any finite distance from the charge. If we move away from the charge, i.e., in the direction of the electric field we move towards the lower values of potential. If we move towards the direction opposite to that of electric field we move towards the higher values of potential.

 

 

The above figure shows the electric field of a point negative charge.

 

 

Finite distance is negative at any point from the charge. Here if we move towards the charge we are moving in the direction of and in the direction of decreasing U (move negative).

If we move away from the charge, in the direction opposite to , we move towards increasing value of V (less negative). Hence, we can conclude that moving with the direction of means moving in the direction of decreasing V and moving against the direction of means moving in the direction of increasing V.

So far, we have expressed potential difference in terms of electric field as

 

 

Now we shall express E as a function of potential as

 

 

where E_r is the component of electric field along the direction of 'r'.

dv/dr is called the potential gradient and the negative sign implies that electric field acts in a direction of decrease of potential.

The above expression also indicates that E is not necessarily zero if V is zero. It is possible to cite examples for cases where E = 0 but V not equal to 0.

For example: The field at the centre of a uniformly charged ring is zero but the potential at the centre is not zero.