The Shortest X-Rays From Thirty Kilovolts

Q. X-rays are produced by 30 kV electrons. What is the minimum wavelength of the X-rays, in nm? expects 0.0413 nm

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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?

The playground

Inputs

voltage

V

30 kV

Outputs

wavelength

λ

The solution

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 Ruby

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

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

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

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