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Chapter 28: Problem 34

\(\cdot\) The cathode-ray tubes that generated the picture in early color televisions were sources of \(x\) rays. If the acceleration voltage in a television tube is 15.0 \(\mathrm{kV}\) , what are the shortest- wavelength \(\mathrm{x}\) rays produced by the television? (Modern televisions contain shielding to stop these x rays.)

Short Answer

Expert verified
The shortest wavelength of X rays produced is 0.0827 nm.

Step by step solution

01

Understanding the problem

We need to find the shortest wavelength of X rays produced by a cathode-ray tube in a television when electrons are accelerated by a voltage of 15.0 kV.
02

Formula for Shortest Wavelength

The energy of the X rays is equal to the energy imparted to the electrons. This energy can be calculated using the formula for the shortest wavelength (also known as the cutoff wavelength) in X-ray production: \( \lambda_{min} = \frac{hc}{eV} \), where \( \lambda_{min} \) is the shortest wavelength, \( h \) is Planck's constant \(6.626 \times 10^{-34} \text{ J s}\), \( c \) is the speed of light \(3.00 \times 10^8 \text{ m/s}\), \( e \) is the charge of an electron \(1.602 \times 10^{-19} \text{ C}\), and \( V \) is the voltage.
03

Substitute Values into the Formula

Substitute the given values into the equation: \( \lambda_{min} = \frac{6.626 \times 10^{-34} \, \text{J s} \times 3.00 \times 10^{8} \, \text{m/s}}{1.602 \times 10^{-19} \, \text{C} \times 15,000 \, \text{V}} \).
04

Calculate the Shortest Wavelength

Perform the calculation: The numerator \( hc = 6.626 \times 10^{-34} \, \text{J s} \times 3.00 \times 10^{8} \, \text{m/s} = 1.988 \times 10^{-25} \, \text{J m} \). The denominator \( eV = 1.602 \times 10^{-19} \, \text{C} \times 15,000 \, \text{V} = 2.403 \times 10^{-15} \, \text{J} \). Divide the results: \( \lambda_{min} = \frac{1.988 \times 10^{-25} \, \text{J m}}{2.403 \times 10^{-15} \, \text{J}} = 8.27 \times 10^{-11} \, \text{m} \).
05

Conclusion

The shortest wavelength \( \lambda_{min} \) of the X rays produced by the television tube with a 15.0 kV acceleration voltage is approximately \( 8.27 \times 10^{-11} \, \text{m} \) or 0.0827 nanometers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

X-ray Production
X-ray production happens when high-energy electrons slam into a metal target. Inside a cathode-ray tube, electrons are accelerated towards this target, releasing X-rays upon impact. This process involves several steps:
  • The cathode emits electrons when heated.
  • An acceleration voltage propels these electrons towards the target.
  • Upon striking the target, the electrons' abrupt deceleration creates X-rays.
These X-rays can vary in wavelength and are an essential tool in both medical imaging and material analysis. However, in older television screens, shielding was necessary to prevent harmful exposure from X-rays produced during operation.
Shortest Wavelength
The shortest wavelength of X-rays, also known as the cutoff wavelength, represents the condition when an electron loses all of its kinetic energy upon impact. To find this wavelength, we use the formula:\[ \lambda_{min} = \frac{hc}{eV} \]Here, \(\lambda_{min}\) is the shortest wavelength, \(h\) is Planck's constant, \(c\) is the speed of light, \(e\) is the electron charge, and \(V\) is the acceleration voltage. This wavelength is critical in determining the limit of the resolution and penetration power of the X-rays produced.
Acceleration Voltage
The acceleration voltage in a cathode-ray tube is what gives electrons their immense speed before they hit the target. This voltage determines the energy of the electrons, and thus, the energy of the X-rays produced. The greater the acceleration voltage:
  • The faster the electrons travel.
  • The more powerful the X-rays produced after impact.
  • The shorter the wavelength of these X-rays.
In the context of older televisions, a typical acceleration voltage was around 15.0 kV. This was enough to produce X-rays with a notable amount of energy, requiring effective shielding to protect users.
Planck's Constant
Planck's constant \(h\) is a key figure in quantum mechanics, defining the scale of energy quantization. With a value of approximately \(6.626 \times 10^{-34} \text{ J s}\), it connects the energy of a photon to its frequency.In X-ray production, Planck's constant helps in determining the minimum wavelength of X-rays produced. It's a critical part of the formula:\[ \lambda_{min} = \frac{hc}{eV} \]The constant indicates that energy and frequency are directly proportional - as the frequency of a photon increases, so does its energy.
Electron Charge
The charge of an electron \(e\) is a fundamental physical constant valued at approximately \(1.602 \times 10^{-19} \text{ C}\). It's the basis for all electric currents and plays a vital role in the behavior of subatomic particles.In the context of X-ray production, the charge of an electron is crucial for understanding how they gain energy from the acceleration voltage:
  • Each electron carries a negative charge as it travels through a voltage potential.
  • When electrons impact the target, they transfer their kinetic energy, producing X-rays.
This constant enables the conversion of electrical energy into X-ray energy, vital for computing the shortest wavelength achievable.

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Most popular questions from this chapter

A radio station broadcasts at a frequency of 92.0 \(\mathrm{MHz}\) with a power output of 50.0 \(\mathrm{kW}\) . (a) What is the energy of each emitted photon, in joules and electron volts? (b) How many photons are emitted per second?

\(\bullet\) The photoelectric threshold wavelength of a tungsten surface is 272 \(\mathrm{nm}\) . (a) What are the threshold frequency and work function (in eV) of this tungsten? (b) Calculate the maximum kinetic energy (in eV) of the electrons ejected from this tungsten surface by ultraviolet radiation of frequency \(1.45 \times 10^{15} \mathrm{Hz}\)

\(\bullet\) The predominant wavelength emitted by an ultraviolet lamp is 248 nm. If the total power emitted at this wavelength is \(12.0 \mathrm{W},\) how many photons are emitted per second?

\(\cdot\) For a hydrogen atom in the ground state, determine, in electron volts, (a) the kinetic energy of the electron, (b) the potential energy, (c) the total energy, (d) the minimum energy required to remove the electron completely from the atom. (e) What wavelength does a photon with the energy calculated in part (d) have? In what region of the electromagnetic spectrum does it lie?

An incident \(x\) -ray photon is scattered from a free electron that is initially at rest. The photon is scattered straight back at an angle of \(180^{\circ}\) from its initial direction. The wavelength of the scattered photon is 0.0830 \(\mathrm{nm.}\) (a) What is the wavelength of the incident photon? (b) What is the magnitude of the momentum of the electron after the collision? (c) What is the kinetic energy of the electron after the collision?
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