Capacitance of a Spherical Conductor

Introduction

A capacitor is a set-up to store electric charge and hence electrostatic potential energy. Capacitors vary in shape and size but the most basic configuration consists of two identical conductors placed close to each other and carrying equal and opposite charges.

A single isolated conductor can also store electric charge (and therefore it has some capacitance) and thus produce electric field in the space around it. This electric field has stored electric potential energy through it. However this energy is distributed or scattered throughout the entire region extending from conductor surface to infinity and since the energy is not confined in a fixed region, it is quite difficult to put it in proper use whenever required in practical situations. Also, the capacitance of a single isolated conductor is generally quite less even when the conductors' size is fairly large.

Capacitance of a Spherical Conductor

As an illustration, we'll consider a spherical conductor carrying some electric charge and try to obtain its capacitance.

When a charge Q is given to an isolated spherical conductor of radius R, then its potential rises.

If we assume the potential at infinity to be zero, potential V of the spherical conductor as a function of charge Q can be written as

Capacitance of the conductor is nothing but the capacity to hold charge per unit voltage i.e. C = Q/V = 4πεᤱR [using above equation].

Thus capacitance of an isolated spherical conductor is given by

Notable Points

Observing the expression of capacitance that we obtained above, following conclusions can be drawn:

• The capacitance of a spherical capacitor does not depend on charge Q and potential V of the conductor. The capacitance is also independent of the energy stored by the charged conductor.
• It depends on the size (radius) of the conductor and also on the surrounding medium.
• Greater the size, larger is the capacitance (i.e. C ∝ R).