Question

Write the relation for the speed of electromagnetic waves in terms of the amplitudes of electric and magnetic.

Answer 1

Hint Use the electric field and magnetic field formula after that use the Maxwell’s equation and after that partially differentiate the electric field and magnetic field equation.

Complete step-by-step solution:In electromagnetic waves ,the ratio of amplitudes of the magnetic field and the electric field is equal to the velocity of the electromagnetic waves in free space.
Let the electric field and magnetic field be constrained to the y and z directions, respectively and they both are functions of only x and t.
Electric field $E=E_m\cos (kx-\omega t)$
Magnetic field $B=B_m\cos (kx-\omega t)$
Where $E=$ electric field, $B=$ magnetic field, $E_m=$ amplitude of electric field, $B_m=$ amplitude of magnetic field, $k=$ wave number and $\omega =$ frequency
Now we use Maxwell’s equation
$\mathrm{\nabla }\times E=-{\displaystyle \frac{\partial B}{\partial t}}$
The curl of the electric field is ${\displaystyle \frac{\partial E}{\partial x}}\hat{k}$
So ${\displaystyle \frac{\partial E}{\partial x}}=-{\displaystyle \frac{\partial B}{\partial t}}$
Now partially differentiating the electric field and magnetic field equations above.
$\Rightarrow k{E}_m\sin (kx-\omega t)=-(-B_m\omega \sin (kx-\omega t))$
$\Rightarrow {\displaystyle \frac{\omega }{k}}={\displaystyle \frac{E_m}{B_m}}$
And $c={\displaystyle \frac{\omega }{k}}$
So $c={\displaystyle \frac{E_m}{B_m}}$
Where $c$ is phase speed of electromagnetic waves.

Note:Student remember that electric field and magnetic field Maxwell’s relation used there and also remember the curl of electric field .Do not forget to do partial differentiation of electric field and magnetic field .