Question

1. In a region of space free of currents, the electric and magnetic fields at t 0 are determined to be: where a is a constant and is a unit vector in the y-direction. (a) What is the charge density at t-0, p(F,0)? (b) Caleulate the magnetic field as a function of tme (c) Calculate the electric field as a function of time

Answer 1

Given, E (r, 0) = alpha x^r j, B (r, 0) = 0 (a) From Maxwell's equation rightarrow nabla E = p/E_0 therefore (i partial differential/ partial differential x + j partial differential/ partial differential y + k partial differential/ partial differential z) (alpha x^r j) = p/E_0 p = E_0 alpha partial differential/ partial differential y (x^2) = 0 (b) nabla times E = -partial differential B/ partial differential t nabla times E = Vertical-bar i partial differential/partial differential x 0 j partial differential/partial differential y alpha x^2 k partial differential/partial differential z 0 Vertical-bar = k (2 alpha x - 0) = 2 alpha x k = -partial differential B/ partial differential t therefore partial differential B/ partial differential t = -2 alpha x k B (r, t) = -2 alpha x t k