A Radio's Shortest Wavelength

solved ElectromagneticWave

Q. A radio can tune in to any station in the 7.5 MHz to 12 MHz $ν = 12$ MHz band. What is the shortest wavelength of the corresponding wavelength band, in metres?

Inputs

frequency

ν

12 MHz

Outputs

wavelength

λ

How it's solved

Solved with ElectromagneticWave

Given

$ν = 12$ MHz

Find $λ$ .

Working

$ν = 12$ MHz $= 1.2 \times 10^{7}$ Hz
light travels at c(wave) $c = ν \cdot λ$
$2.99792458 \times 10^{8} = 1.2 \times 10^{7} \cdot λ$
$λ = 24.98 m$
$λ \approx 25$ mto 2 significant figures

Conclusion

Wavelength and frequency go inversely at the speed of light, so the highest frequency of the band gives its shortest wavelength.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :a_radio_s_shortest_wavelength do4  kind :numerical5  describes "A radio can tune in to any station in the 7.5 MHz to %{frequency} band. %{wavelength}, in metres?"6  given frequency: "12 MHz"7  asks wavelength: "What is the shortest wavelength of the corresponding wavelength band"8end

solution.rb

1require_relative "../questions/a_radio_s_shortest_wavelength"2require_relative "../science/electromagnetic_waves/electromagnetic_wave"34solve :a_radio_s_shortest_wavelength, using: ElectromagneticWave do5  objects :wave67  take frequency: wave.frequency89  wavelength = wave.wavelength10  answer wavelength:, in: "m", figures: 21112  conclusion "Wavelength and frequency go inversely at the speed of light, so the highest frequency of the " \13             "band gives its shortest wavelength."14end

playground.rb

1require_relative "../solutions/a_radio_s_shortest_wavelength"23play_with :a_radio_s_shortest_wavelength do4  title "A Radio's Shortest Wavelength"5end

The laws it drew on

quantity	written	units	within	from
pi	$π$	—	0 … ∞	ElectromagneticWave
speed of light	$c$	m/s	—	ElectromagneticWave
electric field	$E$	V/m	—	ElectromagneticWave
magnetic field	$B$	T	—	ElectromagneticWave
wave number	$k$	m⁻¹	—	ElectromagneticWave
angular frequency	$ω$	rad/s	—	ElectromagneticWave
wavelength	$λ$	m	0 … ∞	ElectromagneticWave
frequency	$ν$	Hz	0 … ∞	ElectromagneticWave
electric field of ground	$E_{g}$	V/m	—
magnetic field of ground	$B_{g}$	T	—
electric field of wave	$E$	V/m	—
magnetic field of wave	$B$	T	—
wave number of ground	$k_{g}$	m⁻¹	—
wavelength of ground	$λ_{g}$	m	0 … ∞
wave number of wave	$k$	m⁻¹	—
wavelength of wave	$λ$	m	0 … ∞
angular frequency of ground	$ω_{g}$	rad/s	—
frequency of ground	$ν_{g}$	Hz	0 … ∞
angular frequency of wave	$ω$	rad/s	—
frequency of wave	$ν$	Hz	0 … ∞

E_{w} = c \cdot B_{w}

fields of a light wave

k_{w} = \frac{2 \cdot π}{λ_{w}}

wave number and wavelength

ω_{w} = 2 \cdot π \cdot ν_{w}

angular frequency and frequency

c = ν_{w} \cdot λ_{w}

light travels at c

E_{g} = c \cdot B_{g}

fields of a light wave(ground)

E = c \cdot B

fields of a light wave(wave)

k_{g} = \frac{2 \cdot π}{λ_{g}}

wave number and wavelength(ground)

k = \frac{2 \cdot π}{λ}

wave number and wavelength(wave)

ω_{g} = 2 \cdot π \cdot ν_{g}

angular frequency and frequency(ground)

ω = 2 \cdot π \cdot ν

angular frequency and frequency(wave)

c = ν_{g} \cdot λ_{g}

light travels at c(ground)

c = ν \cdot λ

light travels at c(wave)