A cone made of insulating material has a total charge Q spread uniformly over its sloping surface having a slant length L . The energy required to take a test charge q from infinity to apex A of the cone will be

Step-by-step solution

σ = Q πRL L = Slant height dQ = σ × 2 πr dx = Charge on the elementary ring dQ = Q πRL × 2 πrdx = Q LR 2 rdx ... (1) By similarity of triangles : R L = r x ⇒ r = R L x ... (2) By (1) and (2) we get dQ = Q LR × 2 R L xdx = 2 Q L 2 xdx ∴ the electric potential at A due to ring is dV = kdQ x = k x 2 Q L 2 xdx ∴ Total potential at A is V = ∫ dv = 2 kQ L 2 ∫ 0 L dx = 2 kQ L 2 × L V = 2 kQ L ∴ Work done to shift a charge q from infinity to point A is = qV W = 2 kqQ L = 2 × 1 4 π ϵ 0 qQ L = qQ 2 π ϵ 0 L