Question

You need to use your cell phone, which broadcasts an 800 MHz signal, but you're behind two massive, radio-wave-absorbing buildings that have only a 15 m space between them. What is the angular width, in degrees, of the electromagnetic wave after it emerges?

Answer 1

Concepts and reason

The concepts needed to solve this problem are single slit diffraction and relation between the frequency, speed, and wavelength of a wave.

Initially, find the wavelength of the broadcasting signal using the frequency of the signal and speed of light. After that find, calculate the half angular width that is diffraction angle $\theta$by using diffraction minimum of single slit diffraction.

To find the angular width of the electromagnetic wave, multiply the angle $\theta$by a factor of two.

Fundamentals

The wavelength of the broadcasting signal is calculated as follows:

$\lambda = \frac{c}{f}$

Here, c is the speed of light and f is the frequency of the broadcasting signal.

The expression for the diffraction minimum in the single slit diffraction is,

$d\sin \theta = m\lambda$

Here, d is the slit width, m is the order of the diffraction, and$\lambda$is the wavelength.

The angular width of the electromagnetic wave that is width of the central maximum is equal to the twice of the angle of diffraction for the first minimum.

$\theta ' = 2\theta$

The wavelength of the broadcasting signal is calculated as follows:

$\lambda = \frac{c}{f}$

Substitute 800 MHz for f and $3.0 \times {10^8}{\rm{ m/s}}$for c in the above expression.

$\begin{array}{c}\\\lambda = \frac{{3.0 \times {{10}^8}{\rm{ m/s}}}}{{800{\rm{ MHz}}\left( {\frac{{{{10}^6}{\rm{ Hz}}}}{{1.0{\rm{ MHz}}}}} \right)}}\\\\ = 0.375{\rm{ m}}\\\end{array}$

Rearrange the equation$d\sin \theta = m\lambda$ for$\theta$.

$\theta = {\sin ^{ - 1}}\left( {\frac{{m\lambda }}{d}} \right)$

The value of m for the first minimum is equal to one.

Substitute 1 for m, 0.375 m for $\lambda$and 15 m for d in the expression$\theta = {\sin ^{ - 1}}\left( {\frac{\lambda }{d}} \right)$.

$\begin{array}{c}\\\theta = {\sin ^{ - 1}}\left( {\frac{{\left( 1 \right)\left( {{\rm{0}}{\rm{.375 m}}} \right)}}{{{\rm{ 15 m}}}}} \right)\\\\ = 1.4^\circ \\\end{array}$

Substitute$1.4^\circ$for$\theta$in the equation$\theta ' = 2\theta$.

$\begin{array}{c}\\\theta ' = 2\theta \\\\ = 2\left( {1.4^\circ } \right)\\\\ = 2.8^\circ \\\end{array}$

Ans:

The angular width of the electromagnetic wave is$2.8^\circ$.