Problem 16 - University Physics with Modern Physics

Question

(a) What is the minimum potential difference between the filament and the target of an x-ray tube if the tube is to produce x rays with a wavelength of 0.150 \(\mathrm{nm}\) (b) What is the shortest wavelength produced in an x-ray tube operated at 30.0 \(\mathrm{kV}\) ?

Step-by-Step Solution

Answer

(a) Minimum potential difference: 8.28 kV. (b) Shortest wavelength: 0.0414 nm.

Step 1: Understand the Relationship

The production of X-rays involves electrons being accelerated by a potential difference (voltage) towards a metal target. The energy of these electrons is converted into X-ray photons. The minimum wavelength of these X-rays can be determined by the maximum energy transferred, which is equal to the energy of the electrons (in electron volts). The relationship between the X-ray photon wavelength \( \lambda \), electron charge \( e \), and potential difference \( V \) is given by the formula \( \lambda = \frac{hc}{eV} \), where \( h \) is Planck's constant and \( c \) is the speed of light.

Step 2: Calculate the Minimum Potential Difference (a)

For the minimum potential difference required to produce X-rays with a wavelength of \( \lambda = 0.150 \, \text{nm} = 0.150 \times 10^{-9} \, \text{m} \), we use the formula mentioned earlier. Rearranging gives \( V = \frac{hc}{e\lambda} \). Substituting the values \( h = 6.63 \times 10^{-34} \, \text{Js} \), \( c = 3 \times 10^8 \, \text{m/s} \), and \( e = 1.6 \times 10^{-19} \, \text{C} \), we find that \( V \approx 8.28 \times 10^3 \, \text{V} \). Therefore, the minimum potential difference is approximately 8.28 kV.

Step 3: Calculate the Shortest Wavelength Produced (b)

Now, to find the shortest wavelength produced when the tube operates at 30.0 kV, we use the same equation \( \lambda = \frac{hc}{eV} \). Substitute \( V = 30.0 \times 10^3 \, \text{V} \), along with the constants for \( h \), \( c \), and \( e \) into the equation. Calculating gives \( \lambda \approx 4.14 \times 10^{-11} \, \text{m} \), or \( \lambda \approx 0.0414 \, \text{nm} \). Hence, the shortest wavelength produced is approximately 0.0414 nm.

Key Concepts

Wavelength

Wavelength is a fundamental concept in the study of light and other electromagnetic waves. It represents the distance between two consecutive peaks (or troughs) of a wave and is typically measured in meters, centimeters, or nanometers for smaller scales like X-rays.
Understanding wavelength is crucial when discussing X-ray production because X-rays are a form of electromagnetic radiation. The specific wavelength of an X-ray determines its energy and, consequently, its ability to penetrate materials.

• Wavelength is inversely proportional to frequency, meaning waves with shorter wavelengths have higher frequencies and energies.
• In X-ray tubes, electrons are accelerated to high speeds to produce X-rays with specific wavelengths.

The ability to calculate the wavelength of X-rays is essential for medical imaging and other applications, as it influences the X-ray's interactions with different materials.

Potential Difference

The potential difference, often referred to as voltage, plays a key role in the process of X-ray production. It is defined as the difference in electric potential between two points and is measured in volts (V).
In the context of X-ray tubes, potential difference is used to accelerate electrons towards a metal target. This acceleration is what generates X-rays.

• The higher the potential difference, the more kinetic energy the electrons gain.
• This increased energy allows electrons to produce X-rays with shorter wavelengths, which are more penetrating.

Understanding potential difference is crucial for determining both the minimum voltage required to produce X-rays with a desired wavelength and the characteristics of the X-rays produced at certain voltages.

Electron Volts

An electron volt (eV) is a unit of energy commonly used in the context of atomic and subatomic processes. It represents the amount of energy gained or lost by an electron when it moves across a potential difference of one volt.
Electron volts are particularly useful in X-ray physics because they provide a convenient measure of the energy of particles like electrons and photons.

• 1 eV is equivalent to approximately 1.6 x 10^-19 joules.
• The concept of electron volts allows for straightforward comparison between the energy of electrons and the X-ray photons they produce.

Understanding electron volts is essential when calculating the energy and implicit characteristics, such as wavelength, of the X-rays generated in an X-ray tube.

Planck's Constant

Planck's constant is a fundamental physical constant denoted by the symbol \( h \). It is pivotal in quantum mechanics, representing the scale at which quantum effects become significant.
In the context of X-ray production, Planck's constant relates to the energy of photons and plays a crucial role in calculating their wavelengths.

• The value of Planck's constant is approximately \( 6.63 \times 10^{-34} \) Js.
• It is used in the formula \( E = hf \), where \( E \) is the energy of a photon and \( f \) is its frequency.

In X-ray tubes, understanding Planck's constant helps in converting the energy of accelerated electrons into the corresponding wavelength of the produced X-ray photons. It is essential for calculating the properties of electromagnetic waves in these high-energy interactions.