A Wave from Its Equation

Q. The magnetic field in a plane electromagnetic wave is given by B_y = (2 × 10⁻⁷ T) sin(0.5 × 10³ rad/m x + 1.5 × 10¹¹ t). What is the wavelength of the wave, in cm? expects 1.26 cm

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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?

The playground

Inputs

wave number

k

500 rad/m

Outputs

wavelength

λ

The solution

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 Ruby

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

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

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

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)