Question

Consider an infinitely long cylinder of radius R with two spherical cavities, also of radius R. The cylinder carries a uniform volume charge density of ρ. There are two point charges at the center of the spherical cavities both of charge q.
Hint: Just as the previous hint, superposition is your friend. A suggestion is to find the contributions from the cylinder and spheres separately.

(a) Find the electric field at the points A, B, and C in the diagram below.
(b) Suppose that the point charges have a magnitude q = 4πR3ρ. Find the energy density of the field outside the cylinder.

3R 2. drawing shows the central crossection of the cylinder 3R 3R

Answer 1

Electric field due to an uniformly charged cylinder: - 1 outside from gauss�s law E_A 2 pi 4 R l = rho pi R^2 l/epsilon_0 E_A = rho R/8 epsilon_0 at (A) & at (B) E_B = rho r/2 epsilon_0 (a) E^rightarrow_A = [rho R/8 epsilon_0 + 2 R (q - rho 4/3 pi R^3) 4 R/(5 R)^3] x^^E^rightarrow_B = rho R/4 epsilon_0 x^^+ 4 q/4 pi epsilon_0 R^2 x^^- rho R/6 epsilon_0 x^^+ R(q - rho 4/3 pi R^3)/r^2[cos theta x^^+ sin theta z^^] here R = 1/4 pi epsilon_0 r = squareroot (6 R)^2 + (R/2)^2 cos theta = R/2 r & sin theta = 6 R/r E^rightarrow_c = O^rightarrow (b) u_E = 1/2 epsilon_0 E^2 [energy density] Electric field outside the cylinder at a distance r from axis of cylinder. q = rho 4 pi R^3 E = rho R^2/2 epsilon r + 2 (rho 4 pi R^3 - rho 4 pi R^3/3) r/4 pi epsilon_0 (9 R^2 + r^2)^3/2 u_E = 1/2 epsilon_0 E^2