Two metal spheres, one of radius R 2 and the other of radius 2 R respectively have the same surface charge density They are brought in contact and separated. The ratio of their new surface charge densities is

Step-by-step solution

Let the surface charge density be . The radii of the two spheres are r₁ = R 2 and r₂ = 2R . The initial charges on the spheres are q₁ = 4 r₁^2 = 4 ( R 2 )^2 = R^2 and q₂ = 4 r₂^2 = 4 (2R)^2 = 16 R^2 . When the spheres are brought into contact, the total charge Q = q₁ + q₂ = 17 R^2 is redistributed such that they reach the same potential V . The new charges q₁' and q₂' satisfy q₁' r₁ = q₂' r₂ , which implies q₁' R/2 = q₂' 2R , so q₂' = 4q₁' . Since q₁' + q₂' = Q , we have 5q₁' = 17 R^2 , so q₁' = 17 5 R^2 and q₂' =