Question

4) (2 x 15 points) Four charges are attached to the corners of a square in thef x, y plane and rotate counterclockwise pr about the center of the square (the z-axis) at the angular frequency w. At the time t-0, the four charges are located as follows: +q at (+a, +a); +q at (-a, +a); -q at (+a, -a); and -q at (-a, -a) a) Calculate the charge density p(x, t) and the current density j(x, t)- pv. b) Calculate the electric dipole moment p(t) /ip(, t) d3x and the magnetic dipole moment m(t)-| ρ( (x,t) d3x

please solve it step by step, because I want understanding the salutation. thanks

Answer 1

Next time you upload an image don't select "auto-rotate"!!

Theta_i (t) = theta_i (0) + wt i = {1, 2, 3, 4} theta_i (0) = pi/4, 3 pi/4, -3 pi/4, -pi/4 x_i (t) = squareroot 2 a Cos theta_i (t) y_i (t) = squareroot 2a Sin theta_i (t) rho_i (x^vector, t) = delta^3 (x - x_i (t), y - y_i (t), z) q_i 3 - 0 dirac delta function. So that rho (x^vector, t) = Sigma^4_i = 1 q_i delta^3 (x - x_i (t), y - y_i (t), z) V^vector = squareroot 2a Cos (-Sin theta_i (t) i^^+ Cos theta_i (t) j^^) j^vector (x, t) = squareroot 2a Cos sigma q_i delta^3 (x - x_i, y - y_i, z) (-Sin theta_i i^^+ Cos theta_i j^^) \r\n x^vector = x i^^+ y j^^ P^vector = integral rho (x^vector, t) d^3 x = integral sigma_i (x i^^+ y j^^) q_i delta^3 (x - x_i, y - y_i, z) d^3x = sigma q_i (x_i i^^+ y_i j^^) P^vector = q_1 (x_1 i^^+ y_i j^^)