A Microwave's Magnetic Field

solved ElectromagneticWave

Q. In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of 2.0 × 10¹⁰ Hz and amplitude 48 V/m $E = 48$ V/m. What is the amplitude of the oscillating magnetic field, in teslas?

Inputs

electric

E

48 V/m

Outputs

magnetic

B

How it's solved

Solved with ElectromagneticWave

Given

$E = 48$ V/m

Find $B$ .

Working

fields of a light wave(wave) $E = c \cdot B$
$48 = 2.99792458 \times 10^{8} \cdot B$
$B = 1.601 \times 10^{−7}$ T
$B \approx 1.6 \times 10^{−7}$ Tto 2 significant figures

Conclusion

The magnetic field of a light wave is its electric field over the speed of light, whatever the frequency.

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_magnetic_field do4  kind :numerical5  describes "In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of 2.0 × " \6            "10¹⁰ Hz and amplitude %{electric}. %{magnetic}, in teslas?"7  given electric: "48 V/m"8  asks magnetic: "What is the amplitude of the oscillating magnetic field"9end

solution.rb

1require_relative "../questions/a_microwave_s_magnetic_field"2require_relative "../science/electromagnetic_waves/electromagnetic_wave"34solve :a_microwave_s_magnetic_field, using: ElectromagneticWave do5  objects :wave67  take electric: wave.electric_field89  magnetic = wave.magnetic_field10  answer magnetic:, in: "T", figures: 21112  conclusion "The magnetic field of a light wave is its electric field over the speed of light, whatever the " \13             "frequency."14end

playground.rb

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