Question

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?

Short answer

The shortest wavelength x-rays produced is approximately \( 8.27 \times 10^{-11} \text{ m} \).

Step-by-step solution

Understand the Concept

To find the shortest wavelength of x-rays produced, we use the equation relating energy and wavelength: \[ E = \frac{hc}{\lambda} \] where \( E \) is energy, \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength.

Convert Voltage to Energy

The energy of the x-ray photon can be directly obtained from the acceleration voltage, since \( E = eV \), where \( e \) is the charge of an electron (\( 1.602 \times 10^{-19} \text{ C} \)), and \( V \) is the voltage. Thus, \[ E = 1.602 \times 10^{-19} \times 15,000 = 2.403 \times 10^{-15} \text{ J} \]

Solve for Wavelength

We rearrange the equation \( \lambda = \frac{hc}{E} \) to solve for the wavelength and then substitute the known values: Planck's constant, \( h = 6.626 \times 10^{-34} \text{ J s} \), and the speed of light, \( c = 3.00 \times 10^8 \text{ m/s} \). Thus, \[ \lambda = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{2.403 \times 10^{-15}} \approx 8.27 \times 10^{-11} \text{ m} \]

Interpret the Results

The calculated wavelength of \( 8.27 \times 10^{-11} \text{ m} \) represents the shortest wavelength, which corresponds to the highest energy x-rays produced by the 15.0 kV acceleration voltage.

Key concepts

Cathode-ray tube

Cathode-ray tubes (CRTs) were once a staple technology in televisions and oscilloscopes. A CRT functions by accelerating electrons toward a screen, creating images when these energetic electrons strike phosphorescent materials. This tube contains a cathode, usually heated to release electrons, and an anode that attracts electrons towards the screen.

The electron beam can be deflected using magnetic or electric fields, allowing it to scan across the screen and generate pictures. X-rays can be produced unintentionally as electrons collide with anode material at high speeds, making it essential for CRTs to have protective shielding to limit x-ray exposure.

Acceleration voltage

Acceleration voltage is crucial in determining the energy of electrons moving through a cathode-ray tube. The voltage, typically measured in kilovolts (kV), accelerates electrons emitted from the cathode to high speeds as they travel toward the screen. A higher acceleration voltage results in higher electron energy.

In the context of x-ray production, the energy of the electrons determines the amount of kinetic energy that gets converted into x-rays when electrons interact with the target material. The greater the voltage, the shorter the wavelength and higher the energy of the generated x-rays. For example, a 15.0 kV acceleration voltage leads to the production of significant x-ray energies.

Planck's constant

Planck's constant ( \( h \) ) is a fundamental constant in physics that plays a critical role in the energy-wavelength relationship of photons. It is valued at approximately \( 6.626 \times 10^{-34} \) Js.

Planck's constant reflects the quantized nature of energy and is foundational in quantum mechanics. When electrons accelerate and emit x-rays in a cathode-ray tube, Planck's constant helps in calculating the energy of these x-rays. This calculation is important for understanding the electromagnetic spectrum and the nature of different types of radiation.

Energy-wavelength equation

The energy-wavelength equation is essential to calculate the characteristics of electromagnetic radiation. The relationship is given by \( E = \frac{hc}{\lambda} \) , where \( E \) is energy, \( h \) is Planck’s constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength.

This equation highlights that energy and wavelength are inversely proportional. As wavelength decreases, the energy increases, explaining why shorter wavelength x-rays are more energetic. In practical terms, the energy provided by an acceleration voltage in a cathode-ray tube can be plugged into this formula to derive the shortest wavelength of x-rays produced.