A Microwave's Wavelength

solved ElectromagneticWave

Q. In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of 2.0 × 10¹⁰ Hz $ν = 2.0 \times 10^{10}$ Hz and amplitude 48 V/m. What is the wavelength of the wave, in cm?

Inputs

frequency

ν

2e+10 Hz

Outputs

wavelength

λ

How it's solved

Solved with ElectromagneticWave

Given

$ν = 2.0 \times 10^{10}$ Hz

Find $λ$ .

Working

light travels at c(wave) $c = ν \cdot λ$
$2.99792458 \times 10^{8} = 2 \times 10^{10} \cdot λ$
$λ = 1.499$ cm
$λ = 0.0149896229$ m $= 1.49896229$ cm
$λ \approx 1.5$ cmto 2 significant figures

Conclusion

At tens of gigahertz the speed of light leaves room for only a centimetre or so per cycle, the microwave part of the spectrum.

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_microwave_s_wavelength do4  kind :numerical5  describes "In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of " \6            "%{frequency} and amplitude 48 V/m. %{wavelength}, in cm?"7  given frequency: "2.0e10 Hz"8  asks wavelength: "What is the wavelength of the wave"9end

solution.rb

1require_relative "../questions/a_microwave_s_wavelength"2require_relative "../science/electromagnetic_waves/electromagnetic_wave"34solve :a_microwave_s_wavelength, using: ElectromagneticWave do5  objects :wave67  take frequency: wave.frequency89  wavelength = wave.wavelength10  answer wavelength:, in: "cm", figures: 21112  conclusion "At tens of gigahertz the speed of light leaves room for only a centimetre or so per cycle, the " \13             "microwave part of the spectrum."14end

playground.rb

1require_relative "../solutions/a_microwave_s_wavelength"23play_with :a_microwave_s_wavelength do4  title "A Microwave's 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)