A plane polarized monochromatic \(EM\) wave is travelling a vacuum along \(z\) direction such that at \(t\, = t_1\) it is found that the electric field is zero at a spatial point \(z_1\) . The next zero that occurs in its neighbourhood is at \(z_2\). The frequency of the electromagnetic wave is:

Given, at \(t=t_{1}, z=z_{1}, E=0\)

the next zero that occurs in it's neighborhood is at \(z_{2},\) the frequency of the electromagnetic wave at \(t_{2}\)

\(e^{i\left(k z_{1}-\omega t_{1}\right)}=e^{i\left(k z_{2}-\omega x_{2}\right)}\)

\(k z_{1}-\omega t_{1}=k z_{2}-\omega t_{2}\)

\(\left(t_{2}-t_{1}\right) \omega=k\left(z-z_{1}\right)\)

where \(k=\frac{2 \pi}{\lambda}=2 \pi v\)

\(\left(t_{2}-t_{1}\right)=\frac{2 \pi}{\lambda \times 2 \pi v}\left(z_{2}-z_{1}\right)\)

\(=\frac{1}{x \times v}\left(z_{2}-z_{1}\right)\)

\(\Rightarrow \lambda \times v=\frac{\left(z_{2}-z_{1}\right)}{\left(t_{2}-t_{1}\right)}=C\)

\(\left(t_{2}-t_{1}\right)=\frac{\left(z_{2}-z_{1}\right)}{C}\)

Frequency is \(f \propto \frac{1}{t}\) then \(\frac{1}{\left(t_{2}-t_{1}\right)}=\frac{C}{\left(z_{2}-z_{1}\right)}\)

\(\therefore \text { Frequency, } f=\frac{3 \times 10^{8}}{\left(z_{2}-z_{1}\right)}\)