The magnetic field component of a plane electromagnetic wave travelling in the…

PHYSICS · Multi-part · CBSE · Physics · Electromagnetic Waves · Electromagnetic Waves

Problem

The magnetic field component of a plane electromagnetic wave travelling in the positive

x

-direction is given by

B(x,t)=B_0\cos(kx-\omega t+\phi).

Consider the information below and answer. (i) Derive expressions for

\dfrac{\partial^{2}B}{\partial x^{2}}

and

\dfrac{\partial^{2}B}{\partial t^{2}}

. Show every differentiation step. (ii) Verify that the function

B(x,t)

satisfies the scalar wave equation in free space only when

\dfrac{\omega}{k}=\dfrac{1}{\sqrt{\mu_{0}\varepsilon_{0}}}

. Clearly state the conclusion. (iii) For an electromagnetic wave, the peak electric and magnetic fields obey

E_0=cB_0

. Using this fact together with the result of part (ii), obtain a final expression that links

E_0

B_0

\omega

and

k