Question

6. Spinning Cylinder A cylinder of radius R and infinite length is made of permanently polarized dielectric. The polarization vector P is everywhere proportional to the radial vector r, such that P = ar, where a is a positive constant. The cylinder rotates around its axis with an angular velocity w This is a non-relativistic problem where wR< c. a) Find the electric field E at a radius r both inside and outside the cylinder. b) Find the magnetic field B at a radius r both inside and outside the cylinder c) What is the total electromagnetic energy stored per unit length of the cylinder before the cylinder started spinning? d) Same as in c) but while it is spinning.

Answer 1

a) The volume charge density Inside the cylinder is p = -nabla p = -nabla (or) = -2a The surface charge density on the cylinder is sigma = n p = e_r (a_r)_r = R = aR the total charge per unit length is therefore - 2a pi R^2 + 2 pi R aR = 0 The total charge per unit length is thus the net total charge of the cylinder is zero from Gauss flux theorem E = integral - a_r/epsilon_0 e_r r < R, r > R b) The volume current density is j = pv = 2a wr c theta and the surface current density is alpha = sigma v = alpha w R^2 e theta if the cylinder is infinitely long by symmetry B = B(r) e_1 nabla times B = mu_0 j n times (B_2 - B_1) = mu_0 alpha here B_1 and B_2 are the magnetic induction inside and outside the cylinder - partial differential B_1/partial differential r = -2 mu_0 a w tau, - partialdiffere