Question

1. As observed by inertial systern S, an electrornagnetic plane wave of angular frequency ω is traveling in the +x direction through free space. Its electric and magnetic fields are given respectively by u(ut) E(z, y, z,t) = E0% cos (kz-at), B(z, y, z,t) =-e, cos(kz-wt) , where Eo is the electric-field amplitude and k-w/c is the magnitude of the propagation vector. Consider an inertial system S' that is moving with velocity Bcé, relative to inertial system S, where 0S B <1. Show that the respective fields observed by system S" are given by where is a constant

Answer 1

Given that: s - frame elementof^vector (x, y, z, t) = elementof_0 e^^_y cos (k x - w t) B^vector (x, y, z, t) = elementof_0 /c e^^_2 cos (k x - w t) velocity if s frame v^vector = beta_c e^^_x E^vector and B^vector are perpendicular to v^vector in s frame, elementof'_1 = gamma (elementof_1 + v times B): gamma = 1/squareroot 1 - beta^2 elementof'_1 = 1/1 - beta^2 [elementof_0 cos (k' x' - w' t')/squareroot 1 - beta^2 e^^_y + beta elementof_0 cos (k' x' - w' t')/squareroot 1 - beta^2 (e^^_x times e^^_z)] elementof'_1 = (1 - beta)/squareroot 1 - beta^2 E_0 cos (k' x' - w' t')/squareroot 1 - beta^2 e^^_y Since 1 - beta^2 = (1 + beta) (1 - beta) E' = squareroot 1 - beta/1 + beta E_0 cos (k' x' - w' t')/squareroot 1 - beta^2 e^^_y elementof' = alpha elementof_0 cos [alpha [k x' - w t'] e^^_y since w = k/c from Doppler effect w' or k' = (1 - beta) x rightarrow (x - vt)/squareroot 1 - beta^2 x