In vacuum, with no charge and no current anywhere, Maxwell's equations are E = 0, B =…

PHYSICS · Subjective · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Maxwell's equations and conservation laws · Electromagnetic waves and their propagation

Electricity, Magnetism and Electromagnetism, Ch Plane electromagnetic waves: polarisation, energy flux and radiation pressure

Problem

In vacuum, with no charge and no current anywhere, Maxwell's equations are

\nabla\cdot\mathbf{E} = 0,\qquad \nabla\cdot\mathbf{B} = 0,\qquad \nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t},\qquad \nabla\times\mathbf{B} = \mu_0\varepsilon_0\frac{\partial\mathbf{E}}{\partial t}.

Consider fields that depend only on

z

and

t

, with the electric field along

y

and the magnetic field along

x

\mathbf{E} = E_y(z,t)\,\hat{\mathbf{y}},\qquad \mathbf{B} = B_x(z,t)\,\hat{\mathbf{x}}.

(a) Show that both divergence equations are satisfied for any functions

E_y(z,t)

and

B_x(z,t)

. (b) Write the two curl equations in Cartesian components and show that only two scalar equations survive; state them. (c) Eliminate

B_x

to obtain a single second-order equation for

E_y

, and state the speed of the waves it describes in terms of

\mu_0

and

\varepsilon_0

. (d) Take

E_y = E_0\cos(kz - \omega t)

with

\omega = k/\sqrt{\mu_0\varepsilon_0}

. Find

B_x(z,t)

(take the constant of integration to be zero) and verify that

\mathbf{E}\times\mathbf{B}

points along

+z

Browse the problem library