ELECTROSTATIC POTENTIAL AND CAPACITANCE

LINE INTEGRAL OF ELECTRIC FIELD: -


If a unit positive charge is displaced by dl in an electric field of intensity E, work done is given by dW = E. dl Line integration of this equation gives the work done in displacing a unit positive charge from P to Q as 

       Q

W = ∫ E . dl

       P

This work depends only on the initial and final positions of the unit charge and not on the path followed by it. Hence, work done in moving a charge along a closed path is equal to zero. Thus electric field like gravitational field is a conservative field. 

ELECTROSTATIC POTENTIAL: -

The work done by the electric field in moving a unit positive electric charge from an arbitrarily selected reference point q, which may be inside or outside the field, to point P is given by

        p

W = ∫ E . dl

       q

For the selected reference point, the value of WP depends only on the position of point P and not on the path followed in going from reference point to point P.

Let q be at infinity. The electric field at infinite distance due to finite charge distribution will be zero. The electric field due to an infinitely long charged plane at infinite distance will not be zero. However, in practice, one cannot have such a charge distribution.

The work done in a direction, opposing the electric field in bringing a unit positive charge from an infinite position to any point in the electric field is called the static electric potential ( V ) at that point.

Its sign is taken as negative as the work done is in a direction opposite to the electric field. Thus, work done in bringing a unit positive charge from infinity to points P and Q will be

                  P                                   Q

V ( P ) = - ∫ E . dl  and V ( Q ) = -  ∫ E . dl

                 ∞               ∞

So therefore

                                Q            P

V ( Q ) - V ( P ) = - ∫ E . dl + ∫ E . dl

                               ∞     ∞

                             Q

                        = - ∫ E . dl

                             P

This equation gives the electric potential of point Q with respect to point P. Its unit is volt ( joule / coulomb ) denoted by V and its dimensional formula is M1L(exp2)T(exp – 3)A(exp – 1).