Question

Electromagnetic Waves Ranking
A) from Fastest to slowest Rank these electromagnetic waves on the basis of their speed (in vacuum).
Rank from fastest to slowest. To rank items as equivalent, overlap them
yellow light ,green lights ,x ray, FM radio wave,AM radio wave,infrared light
B) Rank these electromagnetic waves on the basis of their wavelength.
Rank from longest to shortest. To rank items as equivalent, overlap them.
C) Rank these electromagnetic waves on the basis of their frequency.
Rank from largest to smallest. To rank items as equivalent, overlap them.

Answer 1

Concepts and reason

The concepts used to solve this problem are, speed of the electromagnetic light waves; wavelength and frequency of the electromagnetic light waves

Initially, use the expression for the speed of electromagnetic waves in vacuum and rank the electromagnetic light waves on the basis of these speeds in vacuum. Next, use the properties and values of wavelengths of the electromagnetic light waves to rank the electromagnetic light waves on the basis of their wavelengths.

Finally, use the properties and values of the frequencies of the given electromagnetic light waves to rank these electromagnetic light waves on the basis of their frequencies.

Fundamentals

Write the expression for the speed of electromagnetic waves.

$v = \lambda f$

Here, $\lambda$ is the wavelength of the light wave and $f$ is the frequency of the light wave.

Write the expression for the speed of electromagnetic waves in vacuum.

$v = \frac{1}{{\sqrt {{\mu _o}{\varepsilon _o}} }}$

Here, ${\mu _o}$ is the permeability of vacuum for magnetic field which is equivalent to $4\pi \times {10^{ - 7}}\;{\rm{H/m}}$ , and ${\varepsilon _o}$ is the permeability of vacuum for electric field which is equivalent to $8.854 \times {10^{ - 12}}\;{\rm{F/m}}$ .

Speed of the light is,

$c = 3 \times {10^8}\;{\rm{m/s}}$ .

Yellow light: The yellow light is the electromagnetic light. The wavelength of this light varies from $5.7 \times {10^{ - 7}}\;{\rm{m}}$ to $5.6 \times {10^{ - 7}}\;{\rm{m}}$ . The frequency of the yellow light varies from $5.3 \times {10^{14}}\;{\rm{Hz}}$ to $5.2 \times {10^{14}}\;{\rm{Hz}}$ .

Green light: The green light is the electromagnetic light. The wavelength of this light varies from $5.6 \times {10^{ - 7}}\;{\rm{m}}$ to $4.9 \times {10^{ - 7}}\;{\rm{m}}$ . The frequency of the green light varies from $6.1 \times {10^{14}}\;{\rm{Hz}}$ to $5.3 \times {10^{14}}\;{\rm{Hz}}$ .

X-rays: The X-ray is the electromagnetic ray. The wavelength of this rays varies from $6 \times {10^{ - 12}}\;{\rm{m}}$ to $8 \times {10^{ - 9}}\;{\rm{m}}$ . The frequency of the X-rays varies from $3.4 \times {10^{16}}\;{\rm{Hz}}$ to $5 \times {10^{19}}\;{\rm{Hz}}$ .

FM radio wave: The FM radio wave is the electromagnetic wave. The wavelength of this wave varies from $1 \times {10^0}\;{\rm{m}}$ to $1 \times {10^2}\;{\rm{m}}$ . The frequency of the FM radio wave varies from $8.8 \times {10^7}\;{\rm{Hz}}$ to $10.8 \times {10^7}\;{\rm{Hz}}$ .

AM radio wave: The AM radio wave is the electromagnetic wave. The wavelength of this wave varies from $1 \times {10^2}\;{\rm{m}}$ to $1 \times {10^3}\;{\rm{m}}$ . The frequency of the AM radio wave varies from $5 \times {10^5}\;{\rm{Hz}}$ to $10 \times {10^5}\;{\rm{Hz}}$ .

Infra-red light: The infra-red light is the electromagnetic light. The wavelength of this light varies from $7.6 \times {10^{ - 7}}\;{\rm{m}}$ to $0.001\;{\rm{m}}$ . The frequency of the infra-red light varies from $3 \times {10^{11}}\;{\rm{Hz}}$ to $3.9 \times {10^{14}}\;{\rm{Hz}}$ .

(A)

Consider, ${v_y}$ is the speed of yellow light, ${v_g}$ is the speed of green light, ${v_x}$ is the speed of X-rays, ${v_{FM}}$ is the speed of FM radio wave, ${v_{AM}}$ is the speed of AM radio wave, and ${v_{ir}}$ is the speed of infra-red light.

Write the expression for the speed of electromagnetic waves in vacuum.

$v = \frac{1}{{\sqrt {{\mu _o}{\varepsilon _o}} }}$

Here, ${\mu _o}$ is the permeability of vacuum for magnetic field which is equivalent to $4\pi \times {10^{ - 7}}\;{\rm{H/m}}$ , and ${\varepsilon _o}$ is the permeability of vacuum for electric field which is equivalent to $8.854 \times {10^{ - 12}}\;{\rm{F/m}}$ .

Substitute, $4\pi \times {10^{ - 7}}\;{\rm{H/m}}$ for ${\mu _o}$ and $8.854 \times {10^{ - 12}}\;{\rm{F/m}}$ for ${\varepsilon _o}$ .

$\begin{array}{c}\\v = \frac{1}{{\sqrt {\left( {4\pi \times {{10}^{ - 7}}\;{\rm{H/m}}} \right)\left( {8.854 \times {{10}^{ - 12}}\;{\rm{F/m}}} \right)} }}\\\\ = 3.0 \times {10^8}\;{\rm{H}} \cdot {\rm{F/}}{{\rm{m}}^2}\\\\ = 3.0 \times {10^8}\;{\rm{H}} \cdot {\rm{F/}}{{\rm{m}}^2}\left( {\frac{{\;{{\rm{m}}^3}{\rm{/s}}}}{{\;{\rm{H}} \cdot {\rm{F}}}}} \right)\\\\ = 3.0 \times {10^8}\;{\rm{m/s}}\\\end{array}$

Thus, $v$ is equivalent to the speed of light.

Since, in vacuum, the electromagnetic waves travel with the speed of light, all these electromagnetic waves have same speed.

(B)

Consider, ${\lambda _y}$ is the wavelength of yellow light, ${\lambda _g}$ is the wavelength of green light, ${\lambda _x}$ is the wavelength of X-rays, ${\lambda _{FM}}$ is the wavelength of FM radio wave, ${\lambda _{AM}}$ is the wavelength of AM radio wave, and ${\lambda _{ir}}$ is the wavelength of infra-red light.

Write the values of the wavelengths of yellow light, green light, X-rays, FM radio wave, AM radio wave, and infra-red light.

The wavelength of the yellow light varies from $5.7 \times {10^{ - 7}}\;{\rm{m}}$ to $5.6 \times {10^{ - 7}}\;{\rm{m}}$ .

The wavelength of the green light varies from $5.5 \times {10^{ - 7}}\;{\rm{m}}$ to $4.9 \times {10^{ - 7}}\;{\rm{m}}$ .

The wavelength of the X-rays varies from $6 \times {10^{ - 12}}\;{\rm{m}}$ to $8 \times {10^{ - 9}}\;{\rm{m}}$ .

The wavelength of the FM radio waves varies from $1 \times {10^0}\;{\rm{m}}$ to $1 \times {10^2}\;{\rm{m}}$ .

The wavelength of the AM radio wave varies from $1 \times {10^2}\;{\rm{m}}$ to $1 \times {10^3}\;{\rm{m}}$ .

The wavelength of the infra-red light varies from $7.6 \times {10^{ - 7}}\;{\rm{m}}$ to $0.001\;{\rm{m}}$ .

From the above values for the wavelength of the given electromagnetic light waves, the order of the wavelength from longest to shortest wavelength is written as, ${\lambda _{AM}} > {\lambda _{FM}} > {\lambda _{ir}} > {\lambda _y} > {\lambda _g} > {\lambda _x}$ .

(C)

Consider, ${f_y}$ is the frequency of yellow light, ${f_g}$ is the frequency of green light, ${f_x}$ is the frequency of X-rays, ${f_{FM}}$ is the frequency of FM radio wave, ${f_{AM}}$ is the frequency of AM radio wave, and ${f_{ir}}$ is the frequency of infra-red light.

Write the values of the frequencies of yellow light, green light, X-rays, FM radio wave, AM radio wave, and infra-red light.

The frequency of the yellow light varies from $5.3 \times {10^{14}}\;{\rm{Hz}}$ to $5.2 \times {10^{14}}\;{\rm{Hz}}$ .

The frequency of the green light varies from $6.1 \times {10^{14}}\;{\rm{Hz}}$ to $5.3 \times {10^{14}}\;{\rm{Hz}}$ .

The frequency of the X-rays varies from $3.4 \times {10^{16}}\;{\rm{Hz}}$ to $5 \times {10^{19}}\;{\rm{Hz}}$ .

The frequency of the FM radio waves varies from $8.8 \times {10^7}\;{\rm{Hz}}$ to $10.8 \times {10^7}\;{\rm{Hz}}$ .

The frequency of the AM radio wave varies from $5 \times {10^5}\;{\rm{Hz}}$ to $10 \times {10^5}\;{\rm{Hz}}$ .

The frequency of the infra-red light varies from $3 \times {10^{11}}\;{\rm{Hz}}$ to $3.9 \times {10^{14}}\;{\rm{Hz}}$ .

From the above values of the frequencies of the given electromagnetic light waves, the order of the frequencies from largest to smallest frequency as, ${f_x} > {f_g} > {f_y} > {f_{ir}} > {f_{FM}} > {f_{AM}}$ .

Ans: Part A

The speed of electromagnetic waves in vacuum is ranked as ${v_y} = {v_g} = {v_x} = {v_{FM}} = {v_{AM}} = {v_{ir}}$ .

Part B

The wavelengths of the given electromagnetic waves are in the following order, ${\lambda _{AM}} > {\lambda _{FM}} > {\lambda _{ir}} > {\lambda _y} > {\lambda _g} > {\lambda _x}$ .

Part C

The frequencies of the electromagnetic waves are as ${f_x} > {f_g} > {f_y} > {f_{ir}} > {f_{FM}} > {f_{AM}}$ .