Problem 14 - University Physics with Modern Physics

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 \(\mathrm{kV}\) , what are the shortest-wavelength \(\mathrm{x}\) rays produced by the television? (Modern televisions contain shielding to stop these \(x\) rays.)

Step-by-Step Solution

Answer

The shortest wavelength of the x-rays is approximately \(8.26 \times 10^{-11} \text{ m}\).

Step 1: Understanding the Problem

We are asked to find the shortest-wavelength of the x-rays produced by a television tube when electrons are accelerated through a potential difference of 15.0 kV. This involves using the relationship between energy and wavelength of the x-rays.

Step 2: Using Energy-Wavelength Relation

To find the wavelength, use the equation relating energy and wavelength: \( E = \frac{hc}{\lambda} \). Here, \(E\) is the energy of the x-ray photon, \(h\) 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. Rearranging gives \( \lambda = \frac{hc}{E} \).

Step 3: Calculating the Energy

The energy \(E\) of the x-rays in electron volts (eV) can be calculated from the voltage: \( E = e \times V \), where \(e\) is the elementary charge \((1.602 \times 10^{-19} \text{ C})\), and \(V = 15.0 \text{ kV} = 15,000 \text{ V}\). Therefore, \(E = 1.602 \times 10^{-19} \times 15,000 = 2.403 \times 10^{-15} \text{ J}\).

Step 4: Finding the Shortest Wavelength

Substitute the values into the rearranged formula for wavelength: \(\lambda = \frac{hc}{E}\). This gives \(\lambda = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{2.403 \times 10^{-15}} \approx 8.26 \times 10^{-11} \text{ m}\).

Step 5: Conclusion

Thus, the shortest wavelength of the x-rays produced by the television is approximately \(8.26 \times 10^{-11} \text{ m}\).

Key Concepts

Cathode-Ray Tube

A cathode-ray tube, commonly used in older television sets and computer monitors, is a type of vacuum tube that contains electron guns and a phosphorescent screen. The basic function of a cathode-ray tube is to accelerate and deflect electron beams to create images on the screen. Inside the tube:

• Electron guns at the back of the tube emit electrons.
• The electrons are accelerated and focused into a narrow beam.
• This beam is directed towards a phosphorescent screen at the front of the tube.
• The screen glows when struck by electrons, creating visible images.

Cathode-ray tubes rely on electric fields to accelerate electrons. When these electrons strike certain materials in the screen, they emit visible light or even x-rays. In early televisions, these x-rays were a by-product of the high-energy electron interactions and required special shielding to prevent exposure.

Energy-Wavelength Relation

In physics, the relationship between the energy of an x-ray and its wavelength is crucial for understanding how imaging and radiation devices work. The core equation that describes this relationship is:\[ E = \frac{hc}{\lambda} \]Where:

• \(E\) represents the energy of the photon in joules.
• \(h\) is Planck's constant \(6.626 \times 10^{-34} \text{ J s}\).
• \(c\) is the speed of light \(3.00 \times 10^8 \text{ m/s}\).
• \(\lambda\) is the wavelength in meters.

This equation implies that high energy corresponds to short wavelengths. This relationship is exploited in many technological applications, ranging from medical imaging to understanding atomic structures. In the context of x-rays, it is the high energy of the photons that allows them to penetrate materials, making them useful for imaging.

Electron Acceleration Voltage

Electron acceleration voltage is an important concept in the function of cathode-ray tubes and the production of x-rays. It refers to the potential difference through which electrons are accelerated:

• Electrons are accelerated by an electric field generated by the voltage.
• The greater the voltage, the higher the energy of the accelerated electrons.
• Higher energy electrons can result in the emission of x-rays when they hit a material.

In the case of a television tube with an acceleration voltage of 15.0 kV:

• The electrons gain energy proportional to this voltage, which can be calculated using: \( E = e \times V \).
• Here, \(e = 1.602 \times 10^{-19} \text{ C}\) (elementary charge).

This energy determines the characteristics of the emitted x-rays, such as their wavelength, with higher acceleration voltages typically producing shorter wavelengths. Such control of x-ray production is vital for both safety and functionality in devices that use these tubes.