Electric Charge and Its Nature-Electrical Capacitance

The electrical capacitance of a conductor is the ability of the conductor to store electric charge. Let us assume that charge Q is given to a conductor, V be the potential difference developed and C be the capacitance of the conductor.

Then as per the definition of the electric capacitance of a conductor:

C=Q/V

In other words, the capacity of a conductor can be defined as the ratio of the charge stored in the conductor to the potential difference developed across the conductor.

Putting V=1, we have C=Q. Hence the capacitance of the conductor is defined as the quantity of charge stored when a potential difference is developed between the conductors.

Units and dimensions

The SI unit of electric capacitance is Farad (F).

1 Farad=1 Coulomb/1 Volt

A conductor is said to have a capacity of 1 Farad when the charge of one coulomb raises its potential by 1 volt. 

The capacitance of some standard capacitors

Spherical capacitor

Let r be the radius of a spherical capacitor, q be the charge contained in it and V be the potential of the conductor.

The potential of the conductor is defined as :

v=1/(4πεr)

C=q/V

With the values, we get C=4πεr

This is the required expression for finding the electric capacitance of a spherical conductor.

Parallel plate capacitor

Let 1 and 2 be the two plates each of area A and distance between them =d.

The electric field between the plates is given by,

E=σ/ε=Q/εA

The potential difference between the plates is given by,

V=E*d=Q/εA *d

C=Q/V

Putting the values in the equation, we get:

C=Aε/d

This is the expression to find the electric capacitance of a parallel plate capacitor.

Electric Capacitance when the medium is present

Let the capacitance of a conductor be C0. If a medium of dielectric constant k is placed in the capacitor, then the electric capacitance of the capacitor changes K times as follows:

C=K*C0

So the electric capacitance depends on the medium.

Energy stored in a capacitor

Let us assume Q is the charge stored in a capacitor and V is the potential difference induced due to the presence of charge.

So as per the definition Q=CV

Also, the work done due to this configuration is W=QV

Let us assume a battery to deliver an infinitely small charge, say DQ, at constant potential V. Then the work done is:

dw=Vdq

dw=q/C DQ

On integrating the above equation from 0 to Q, we get

U=1/2*Q2/C

We know that Q=CV substituting the values in the above equation we get:

U=1/2 CV2

Feature of energy stored in the capacitor

1. The energy stored is independent of the configuration of charges present inside the capacitor.

2. The potential energy is stored in the electric medium between the plates of the capacitor.

3. The potential energy of a capacitor is obtained from the chemical energy stored in the batteries.

The total energy stored in a combination of capacitors

The total energy stored in a period or a parallel combination of capacitors is equal to the sum of the individual energy stored in each capacitor.

Mathematically let U1, U2, U3……………………….. Un be n capacitors and U be the equivalent capacitor, then

U=U1+U2+U3+……………………..Un

Energy density

The energy density is defined as the total energy stored per unit volume of the capacitor. 

Let u be the energy density and U be the total energy v be the volume.

Then energy density=U/v

U= ½ CV2

V=Ad

E=V/d

Substituting the values we get:

U=½ εE2

E is the electric field inside the conductor.

Conclusion

The electric capacitance of any conductor is the ability of the conductor to store electric charge in it. The formula connecting electric capacitance, potential and the charge stored in it is Q=C/V. The electric capacitance can also be defined as the ratio of the charge stored inside it and the potential difference induced. 

The unit of the capacitance of any conductor is the Farad (F). A conductor is said to have the capacity of 1 Farad, when the charge of 14 mm radius is potential by 1 volt. The larger the radius of a spherical conductor, the greater will be its capacitance. The capacitance of a conductor depends on its shape, size and configuration of charge. It also depends on the nature of the medium present inside the capacitor, which is determined through the dielectric constant denoted by K.