Question

2 BALL AND PLANE Consider a spherical shell of radius R and charge per unit area ơi sitting at the origin. There is also an infinite plane parallel to the x - y plane sitting at zzo with charge per unit area σ2. We will take z02R. Compute the electric field at the following locations: 2.1 10 POINTS The origin. 2.2 15 POINTS The point (xo.0,0) with xo>R 2.3 15 POINTS The point (x1,0, z) with 0 < z.< zo and xi> R.

2.1 2.2 2.3

Answer 1

The origin origin lies inside the spherical shell so electronic field due to shell at origin = 0 only electric field will be due to infinite plane E_o^vector = sigma_2/2 epsilon_o (-k^) 2.2 (x, 0, 0) E_shell = k Q_shell/x_o^2 = k (sigma_1 4 pi R^2/x_o^2) = sigma_1 R^2/epsilon_o x_o^2 E_plane = sigma_2/2 epsilon_o E_net^vector (x, 0, 0) = (sigma_1 R^2/epsilon_o x_o^2) i^+ (sigma_2/2 epsilon_o) (-k^) (x_1, 0, z_1) E_shell = sigma_1 R^2/epsilon (x_1^2 + z_1^2) E_plane = sigma_2/2 epsilon_o \r\n in vector form: E^vector_shell = sigma_1 R^2/epsilon_o (x_1^2 + z_1^2) (cos theta i^+ sin theta j^) = sigma_1 R^2/epsilon_o (x_1^2 + z_1^2) [x_1/squareroot x_1^2 + z_1^2 i^+ z_1/squareroot x_1^2 + z_1^2 j^] therefore E_net^vector_(x, 0, z) = (sigma_1 R^2 x_1)/epsilon_o (x_1^2 + z_1^2)^3/2)i^+ (sigma_1 R^2 z_1/epsilon_o(x_1^2 + z_1^2)^3/2 - sigma_2/2 epsilon_o)k^