Question

A sphere of radius R surrounds a particle with charge Q located at its center as shown in figure below. Find the electric flux through a circlular cap of half-angle θ. ) (1-2cos0) 2Eo

Answer 1

Consider sphere of radius R Surrounds particle of charge Q now total flux = E middot A so phi_E = E middot dA E = kQ/R^2 and dA = integral^Q_0 d(dA) d(dA) = 2 pi r times dr sin (phi) = r/R rightarrow r = R sin phi and dr = Rd phi so d(dA) = 2 pi r times dr = 2 pi R sin phi R d phi = 2 pi R^2 sin phi d phi dA = integral^Q_0 d(dA) = 2 pi R^2 [cos phi]^theta_0 = 2pi R^2 [1 - cos theta] hence: phi_E = kQ/R^2 middot 2pi R^2 [1 - cos theta] = Q/R^2 pi epsilon_0 middot 2 pi R^2 (1 -cos theta) phi_E = Q/2 epsilon_0 (1 - cos theta)