Question

A plane polarized monochromatic EM waves is travelling in vacuum along z direction such that t = t1 it is found that the electric field is zero at a spatial point z1. The next zero occurs in its neighborhood is at z2. The frequency of the electromagnetic wave is
A. $\dfrac{{3 \times {{10}^8}}}{{|{z_2} - {z_1}|}}$
B. $\dfrac{{1.5 \times {{10}^8}}}{{|{z_2} - {z_1}|}}$
C. $\dfrac{{6 \times {{10}^8}}}{{|{z_2} - {z_1}|}}$
D. $\dfrac{1}{{{t_1} + \dfrac{{|{z_2} - {z_1}|}}{{3 \times {{10}^8}}}}}$

Answer 1

Hint: Before attempting this question one should have prior knowledge about the electromagnetic wave and the electric field equation for electromagnetic waves also remember to equate electric field at z1 and electric field at z2, use this information to approach towards the solution.

According to the given information a monochromatic EM waves is travelling along the z axis whose electric field are zero at point z1 and z2 it is also given that t = t1
For t = t1 at point z1 electric field is zero i.e. E = 0
Let for t = t2 electric field is zero i.e. E = 0 at point z2
As we know that the general equation of the electric field for an electromagnetic wave is given by $E = {E_o}{e^{ - \left( {kz - \omega t} \right)}}$
Substituting the given values in the above equation for both t1 and t2 and equating them
${E_o}{e^{ - \left( {k{z_1} - \omega {t_1}} \right)}} = {E_o}{e^{ - \left( {k{z_2} - \omega {t_2}} \right)}}$
$ \Rightarrow $\[{e^{ - \left( {k{z_1} - \omega {t_1}} \right)}} = {e^{ - \left( {k{z_2} - \omega {t_2}} \right)}}\]
$ \Rightarrow $\[k{z_1} - \omega {t_1} = k{z_2} - \omega {t_2}\]
$ \Rightarrow $\[k{z_1} - k{z_2} = \omega {t_1} - \omega {t_2}\]
$ \Rightarrow $\[k\left( {{z_1} - {z_2}} \right) = \omega \left( {{t_1} - {t_2}} \right)\]
$ \Rightarrow $\[{z_1} - {z_2} = \dfrac{\omega }{k}\left( {{t_1} - {t_2}} \right)\]
Since we know that $\omega = 2\pi f$ and K is the constant whose value is $k = \dfrac{{2\pi }}{\lambda }$
Substituting the given values in the above equation we get
\[{z_1} - {z_2} = \dfrac{{2\pi f\lambda }}{{2\pi }}\left( {{t_1} - {t_2}} \right)\]
$ \Rightarrow $\[{z_1} - {z_2} = f\lambda \left( {{t_1} - {t_2}} \right)\]
We know that $C = f\lambda $
Therefore \[{z_1} - {z_2} = C\left( {{t_1} - {t_2}} \right)\]
We know that frequency $f$ inversely proportional to $\dfrac{1}{t}$
Therefore \[{z_1} - {z_2} = C\dfrac{1}{f}\]
$ \Rightarrow $\[f = \dfrac{C}{{{z_1} - {z_2}}}\]
Substituting the given value of C in the above equation
\[f = \dfrac{{3 \times {{10}^8}}}{{{z_1} - {z_2}}}\]
Therefore the frequency of the electromagnetic wave is \[\dfrac{{3 \times {{10}^8}}}{{{z_1} - {z_2}}}\]
Hence option A is the correct option.

Note: The concept of electromagnetic waves can be explained as the waves which are formed by the combination in such a way that the electric field and magnetic field are perpendicular to each other, the electromagnetic shows the transverse nature, for the formation of electromagnetic waves accelerated charges are responsible and these types of waves doesn’t require any medium to propagate.