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 $15.0$ kV, what are the shortest-wavelength x rays produced by the television?

Verified step by step guidance

1. Understand that the shortest-wavelength x-rays are produced when electrons are accelerated to their maximum energy and then suddenly stopped, converting their kinetic energy into electromagnetic radiation.
2. Recall the formula that relates the energy of the x-rays to their wavelength: \( E = \frac{hc}{\lambda} \), where \( E \) is the energy, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ J s}) \), \( c \) is the speed of light \( (3.00 \times 10^8 \text{ m/s}) \), and \( \lambda \) is the wavelength.
3. Convert the acceleration voltage from kilovolts to joules. Since 1 electron volt (eV) is equal to \( 1.602 \times 10^{-19} \text{ J} \), multiply the voltage by this conversion factor: \( E = 15,000 \text{ eV} \times 1.602 \times 10^{-19} \text{ J/eV} \).
4. Substitute the energy \( E \) into the formula \( \lambda = \frac{hc}{E} \) to find the wavelength. This will give you the shortest wavelength of the x-rays produced.
5. Ensure units are consistent and perform the calculation to find \( \lambda \), which will be in meters. This is the shortest wavelength of x-rays produced by the television tube.

Key Concepts

Cathode-Ray Tube

A cathode-ray tube (CRT) is a vacuum tube containing one or more electron guns and a phosphorescent screen used to display images. Electrons are accelerated and deflected by electric and magnetic fields to create images on the screen. In early color televisions, CRTs were used to generate images, and the high voltage applied to accelerate electrons could produce x-rays.

X-ray Production

X-rays are produced when high-energy electrons strike a material, typically a metal target. The sudden deceleration of electrons results in the emission of x-rays, a form of electromagnetic radiation. The energy and wavelength of the x-rays depend on the acceleration voltage applied to the electrons; higher voltages produce shorter wavelengths and more energetic x-rays.

Energy-Wavelength Relationship

The energy of a photon is inversely proportional to its wavelength, as described by the equation E = hc/λ, where E is the energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. In the context of x-ray production, the maximum energy of the x-rays corresponds to the shortest wavelength, determined by the acceleration voltage applied to the electrons.