Capacitance of a Cylindrical Capacitor

Consider a cylindrical capacitor formed by two coaxial cylinders A and B of radii a and b respectively and each of length l. Air is the medium between A and B. The outer cylinder B is earthed.



If a charge + q is given to the inner cylinder, then an equal charge – q is induced on the inner surface of the outer cylinder and a charge +q on the outer surface of the outer cylinder. The charge +q induced on the outer surface of the outer cylinder flows to the earth.

The electric field at a point P in the space between the two cylinders at a distance r from the axis is
    E = ___1____ q.            …. (1)
              2πε₀l     r

The potential difference V between the cylinders A and B is
    V = - ∫a E . dl                     ... (2)

Here, d l is the vector displacement along a path from B to A.
Now, E is radially outward and dl is inward. Therefore

    E . dl = E dl cos 180 = - E dl.

As we move a distance  dl from B to A,  we move in the direction of decreasing r. So dl = - dr. Thus

    E . dl = E dr.

Eq (2) becomes, V = - ∫a E dr

         = - ___q___  ∫a dr
                2πε₀l        __ r

    = - ___q___  {log_e r] = - ____q____ {log_e a – log_e b}
        __ 2πε₀l      ______          2πε₀l

= ___q___ log_e  b
      2πε₀l        __  a

Hence the capacitance of the cylindrical capacitor is

C = q = 2πε₀l
       V    log_e (b/a)