A uniform plane electric field along the -direction in free space is described by…

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

Problem

A uniform plane electric field along the

y

-direction in free space is described by

E(x,t)=E_0\sin(kx-\omega t).

Answer the following. (i) Starting with the expression above, obtain the second spatial derivative

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

and the second temporal derivative

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

. Show each step of differentiation clearly. (ii) Using your results from part (i), prove that

E(x,t)

satisfies the three-dimensional vector wave equation

\nabla^{2}E=\mu_{0}\varepsilon_{0}\,\dfrac{\partial^{2}E}{\partial t^{2}}

provided a specific relation holds between

\omega

and

k

. (iii) Hence obtain the condition relating

\omega

and

k

for a plane electromagnetic wave propagating in free space and name the physical quantity represented by the ratio

\dfrac{\omega}{k}

. (iv) If the wavelength of the wave is

500\,\text{nm}

, calculate its frequency.