Let us consider two conducting spheres A and B radii `r_1` and `r_2` repectively connected to each other by a this conducting wire as shown in the figure. The distance between the spheres is much greater than the radii of either spheres.
If a charge Q is introduced into any one of the spheres, this charge Q is redistributed into both the spheres such that tha electrostatic potential is same in both the spheres. They are now uniformaly charged and attain electrostatic equilibrium. Let `q_1` be the charge residing on the surface A and `q_2` is the charge residing on the surface of sphere B such that `Q = q_1 + q_2`. the charges is no net charge inside the conductor. The electrostatic potential at the surface of the sphere A is given by
`V_A = 1/(4piepsilon_0)q_1/r_1` ...(1)
The electrostatic potential at the surface of the sphere B is given by
`V_B = 1/(4piepsilon_0)q_2/r_2` ...(2)
The surface of the conductor is an equipotential. Since the spheres are connected by the conducting wire, the surfaces of both the spheres together form an equipotential surface. This impiles that
`V_A = V_B`
(or) `q_1/r_1 = q_2/r_2` ...(3)
Let us take the charge density on the surface of sphere A is `sigma_1` and charge density on the surface of sphere B is `sigma_2`. This implies that `q_1 = 4pir_1^2sigma_1` and `q_2 = 4pir_2^2sigma_2`, substituting these values into equation (3) we get,
`sigma_1r_1 = sigma_2r_2` ...(4)
from which we conclude that
`sigmar` = constant ...(5)
Thus the surface charge density is inversely proportional to the radius of the sphere. For a smaller radius, the charge density will be larger and vice versa.
Lightning arrester or lightning conductor: It works on the principle of action at points or corona discharge.
This device consists of a long thick copper rod passing from top of the building to the ground. The upper end of the rod has a sharp spike or a sharp needle.
The lower end ofthe rod is connected to the copper plate which is buried deep into the ground. When a negatively charged cloud is passing above the building, a positive charge is induced on the spike. Since the induced charge density on thick sharp spike is large, it results in a corona discharge. This positive charge ionizes the surrounding air which in turn neutralizes the negative charge in the cloud. The negative charge pushed to the spikes passes through the copper rod and is safely diverted to the earth. The lightning arrester does not stop the lightning, eather it diverts the lightning to the ground safely.
