Electrostatic Potential and Capacitance Test - 21

The direction of electric field is always perpendicular to the equipotential surface maintained at high electrostatic potential to other equipotential surface maintained at low electrostatic potential. The positively charged particle experiences the electrostatic force in the direction of electric field i.e., from high electrostatic potential to low electrostatic potential. Thus, the work is done by the electric field on the positive charge, so electrostatic potential energy of the positive charge decreases because speed of charged particle increases which moves in the direction of field due to force \(q\vec E\).

As we know that the relation between electric field intensity E and electric potential V is

\(E=-{dV\over dr}\)

Electric field intensity E = 0 then \(dV\over dr\) = 0

This imply that V = constant

Thus, E = 0 inside the charged conducting sphere then the constant electrostatic potential 100V at every where inside the sphere and it verifies the shielding effect also.

Here we have to findout the shape of equipotential surface, these surface are perpendicular to the field lines, so there must be electric field which can not be without charge. So, the collection of charges, whose total sum is not zero, with regard to great distance can be considered as a point charge. The equipotentials due to point charge are spherical in shape as electric potential due to point charge q is given by

\(V=K_e{q\over r}\)

This suggest that electric potentials due to point charge is same for all equidistant points. The locus of these equidistant points which are at same potential, form spherical surface.

The lines of field from point charge are radial. So the equipotential surface perpendicular to field lines form a sphere.

The direction of electric field is perpendicular to the equipotential surfaces. Here the electric field is in +z direction. And the electric field is always remain in the direction in which the potential decreases. Its magnitude is given by the change in the magnitude of potential per unit displacement normal to the equipotential surface at the point.

So, electric field in z-direction suggests that equipotential surfaces will be the plane perpendicular to z-axis along x-y plane. Therefore the potential is a constant for any x for a given z and for any y for a given z and on the x-y plane for a given z.

In equipotential surface, at any two point, potential difference is zero or equal potential. As we know that the relation between the electric field intensity E and electric potential v is:

\(E=-{dv\over dr}\)

The electric field intensity E is inversely proportional to the separation between equipotential surfaces. So, equipotential surfaces are closer in regions of large electric fields. The electric field intensity is larger near to sharp edges of charged conductor due to larger charge

densities, \(\sigma=({q\over A})\)

As A is very small.

And the electric field E = \(Kq\over r^2\)

So, potential or field decreases as size of body increase or (vice-versa). Hence the equipotential surface will more crowded if charge density (\(\sigma\)) increases.

Work done in displacing a charge particle is given by \(W_{12} = q(V_2 - V_1)\) and the line integral of electrical field from point 1 to 2 gives potential

difference\((V_2-V_1)\) = \(-\int\limits_1^2 E.dl\)

The charge q is moving in the electric field then

work done w = \(-q\int\limits_1^2 (E).dl\)

For the potential on equipotential surface,\((V_2-V_1)\) = 0. Hence, work done on moving a charge is zero (w = 0).

As we know, the relation between electric field intensity E and electric potential V is \(E={-dV\over dr}\) as E = 0 and for V = constant.

\(dV\over dr\) = 0

Thus electric field intensity E = 0.

The main concept used are:

1. The charge resides on the outer surface of a closed charged conductor.

2. Net charge inside the conductor is zero. The excess charge can reside only on the surface of conductor and inside net positive and negative charge is zero, so no any charge can reside inside the hollow shell or body. So, inside the solid material of conducting body there is no charge, it comes to outer surface.