A Wave from Its Equation

solved ElectromagneticWave

Q. The magnetic field in a plane electromagnetic wave is given by B_y = (2 × 10⁻⁷ T) sin(0.5 × 10³ rad/m $k = 0.5 \times 10^{3}$ rad/m x + 1.5 × 10¹¹ t). What is the wavelength of the wave, in cm?

Inputs

wave number

k

500 rad/m

Outputs

wavelength

λ

How it's solved

Solved with ElectromagneticWave

Given

$k = 0.5 \times 10^{3}$ rad/m

Find $λ$ .

Working

$k = 0.5 \times 10^{3}$ rad/m $= 500$ m⁻¹
wave number and wavelength(wave) $k = \frac{2 \cdot π}{λ}$
$500 = \frac{2 \cdot 3.14159265}{λ}$
$λ = 1.257$ cm
$λ = 0.0125663706$ m $= 1.25663706$ cm
$λ \approx 1.26$ cmto 3 significant figures

Conclusion

The number multiplying x counts radians of phase per metre; one wavelength is a full turn.

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_wave_from_its_equation do4  kind :numerical5  describes "The magnetic field in a plane electromagnetic wave is given by B_y = (2 × 10⁻⁷ T) sin(%{wave_number} x + " \6            "1.5 × 10¹¹ t). %{wavelength}, in cm?"7  given wave_number: "0.5e3 rad/m"8  asks wavelength: "What is the wavelength of the wave"9end

solution.rb

1require_relative "../questions/a_wave_from_its_equation"2require_relative "../science/electromagnetic_waves/electromagnetic_wave"34solve :a_wave_from_its_equation, using: ElectromagneticWave do5  objects :wave67  take wave_number: wave.wave_number89  wavelength = wave.wavelength10  answer wavelength:, in: "cm", figures: 31112  conclusion "The number multiplying x counts radians of phase per metre; one wavelength is a full turn."13end

playground.rb

1require_relative "../solutions/a_wave_from_its_equation"23play_with :a_wave_from_its_equation do4  title "A Wave from Its Equation"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)