The Shortest X-Rays From Thirty Kilovolts

solved Photon

Q. X-rays are produced by $E = e \cdot V$ 30 kV $V = 30$ kV electrons. What is the minimum wavelength of the X-rays, in nm?

Inputs

voltage

V

30 kV

Outputs

wavelength

λ

How it's solved

Solved with Photon

Given

$V = 30$ kV
$E = e \cdot V$ from “produced by”

Find $λ$ .

Working

$V = 30$ kV $= 30000$ V
produced by $E = e \cdot V$
$E = 1.60217663 \times 10^{−19} \cdot 30000$
$E = 4.8065299 \times 10^{−15}$ J
energy of a photon(x rays) $E = h \cdot ν$
$4.8065299 \times 10^{−15} = 6.62607015 \times 10^{−34} \cdot ν$
$ν = 7.25396773 \times 10^{18}$ Hz
frequency and wavelength of light(x rays) $c = ν \cdot λ$
$2.99792458 \times 10^{8} = 7.25396773 \times 10^{18} \cdot λ$
$λ = 0.04133$ nm
$λ = 4.13280661 \times 10^{−11}$ m $= 0.0413280661$ nm
$λ \approx 0.0413$ nmto 3 significant figures

Conclusion

A photon that takes an electron's whole energy has the highest frequency, and so the shortest wavelength, the X-rays can have.

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 :the_shortest_x_rays_from_thirty_kilovolts do4  kind :numerical5  describes "X-rays are produced by %{voltage} electrons. %{wavelength}, in nm?"6  given voltage: "30 kV"7  asks wavelength: "What is the minimum wavelength of the X-rays"8end

solution.rb

1require_relative "../questions/the_shortest_x_rays_from_thirty_kilovolts"2require_relative "../science/dual_nature_of_radiation_and_matter/photon"34solve :the_shortest_x_rays_from_thirty_kilovolts, using: Photon do5  objects :x_rays67  take voltage: x_rays.accelerating_potential89  deduce_from "produced by", x_rays.photon_energy => e * x_rays.accelerating_potential1011  wavelength = x_rays.wavelength12  answer wavelength:, in: "nm", figures: 31314  conclusion "A photon that takes an electron's whole energy has the highest frequency, and so the shortest " \15             "wavelength, the X-rays can have."16end

playground.rb

1require_relative "../solutions/the_shortest_x_rays_from_thirty_kilovolts"23play_with :the_shortest_x_rays_from_thirty_kilovolts do4  title "The Shortest X-Rays From Thirty Kilovolts"5end

The laws it drew on

quantity	written	units	within	from
plancks constant	$h$	J s	—	Photon
speed of light	$c$	m/s	—	Photon
elementary charge	$e$	C	—	Photon
frequency	$ν$	Hz	0 … ∞	Photon
wavelength	$λ$	m	0 … ∞	Photon
photon energy	$E$	J	—	Photon
photon momentum	$p$	kg m/s	—	Photon
emitted power	$P$	W	—	Photon
photons a second	$N$	s⁻¹	—	Photon
accelerating potential	$V$	V	—	Photon
photon energy of ground	$E_{g}$	J	—
frequency of ground	$ν_{g}$	Hz	0 … ∞
photon energy of x rays	$E$	J	—
frequency of x rays	$ν$	Hz	0 … ∞
wavelength of ground	$λ_{g}$	m	0 … ∞
wavelength of x rays	$λ$	m	0 … ∞
photon momentum of ground	$p_{g}$	kg m/s	—
photon momentum of x rays	$p$	kg m/s	—
emitted power of ground	$P_{g}$	W	—
photons a second of ground	$N_{g}$	s⁻¹	—
emitted power of x rays	$P$	W	—
photons a second of x rays	$N$	s⁻¹	—
accelerating potential of x rays	$V$	V	—

E_{l} = h \cdot ν_{l}

energy of a photon

c = ν_{l} \cdot λ_{l}

frequency and wavelength of light

p_{l} = \frac{h}{λ_{l}}

momentum of a photon

P_{l} = N_{l} \cdot E_{l}

photons carry the power

E_{g} = h \cdot ν_{g}

energy of a photon(ground)

E = h \cdot ν

energy of a photon(x rays)

c = ν_{g} \cdot λ_{g}

frequency and wavelength of light(ground)

c = ν \cdot λ

frequency and wavelength of light(x rays)

p_{g} = \frac{h}{λ_{g}}

momentum of a photon(ground)

p = \frac{h}{λ}

momentum of a photon(x rays)

P_{g} = N_{g} \cdot E_{g}

photons carry the power(ground)

P = N \cdot E

photons carry the power(x rays)

E = e \cdot V

produced by