Question

Two charged spherical conductors of radii $R_{1}$ and $R_{2}$ when connected by a conducting wire acquire charges q₁ and q₂ respectively. Find the ratio of their surface charge densities in terms of their radius.

Answer 1

Two charged conducting sphere of radii r₁ and r₂ respectively connected to each other by a wire.

so, capacitance of 1st sphere , C₁ = 4πε₀r₁

capacitance of 2nd sphere , C₂ = 4πε₀r₂

After connecting , transfer of charge from higher potential sphere to lower potential sphere and finally, potential of both sphere will be equal .Let final potential is V

Now, charge at 1st sphere, Q₁= C₁V = 4πε₀r₁V

charge at 2nd sphere ,Q₂ = C₂V = 4πε₀r₂V

so, ratio of charge = Q₁/Q₂= 4πε₀r₁V/4πε₀r₂V = r₁/r₂ -------(1)

Now, ratio of electric field = E₁/E₂ = KQ₁/r₁²/KQ₂/r₂² = Q₁r₂²/Q₂r₁²

E₁/E₂ = {Q₁/Q₂}{r₂²/r₁²}

Put equation (1)

E₁/E₂ = r₂/r₁