Question

An electromagnetic standing wave in air has frequency 75.0 MHz. What is the distance between a nodal plane of $\overrightarrow{E}$and the closest nodal plane of $\overrightarrow{B}$?

Short answer

The distance between a nodal plane of $\overrightarrow{E}$and the closest nodal plane of $\overrightarrow{B}$is $1\mathrm{m}$.

Step-by-step solution

Step 1: The standing EM wave

The EM waves can form standing waves when EM waves are reflected from the end of a transmission line.

The wavelength of standing wave produced due to a nodal plane of $\overrightarrow{E}$and nodal plane of$\overrightarrow{B}$

The wavelength of standing wave is produced due to a nodal plane of $\overrightarrow{E}$and nodal plane of $\overrightarrow{B}$is given by

role="math" localid="1664426475142" $\mathrm{x}=\frac{\mathrm{\lambda }}{4}$

Where, $\mathrm{x}$is the distance between two nodal planes and $\mathrm{\lambda }$is the wavelength of wave.

Speed of propagating wave

The speed of propagating wave is given by

$\mathrm{v}=\mathrm{f\lambda }$

Where, $\mathrm{v}$is the speed of propagation of wave and role="math" localid="1664426617586" $\mathrm{f}$is the frequency of wave.

The calculation of wavelength of wave

Given: $\mathrm{f}=75\mathrm{MHz}=75\times {10}^6\mathrm{Hz},\mathrm{v}=3\times {10}^8\mathrm{m}/\mathrm{s}$,

Using

$\mathrm{v}=\mathrm{f\lambda }$

Put the values of constants in above equation

$$\begin{gathered} 3\times {10}^8{\mathrm{ms}}^{-1}=75\times {10}^6\mathrm{Hz}\times \mathrm{\lambda } \\ \mathrm{\lambda }=\frac{3\times {10}^8}{75\times {10}^6} \\ \mathrm{\lambda }=4\mathrm{m} \end{gathered}$$

Thus, the wavelength of wave is $4\mathrm{m}$

The calculation of nodal plane of E→ and closest nodal plane of  B→

Using

$\mathrm{x}=\frac{\mathrm{\lambda }}{4}$

Now, put the value of wavelength in above equation

$$\begin{gathered} \mathrm{x}=\frac{4\mathrm{m}}{4} \\ \mathrm{x}=1\mathrm{m} \end{gathered}$$

Thus, the distance between a nodal plane of $\overrightarrow{E}$and the closest nodal plane oflocalid="1664427274713" $\overrightarrow{B}$is localid="1664427133460" $1\mathrm{m}$.