Question

An X-ray tube is operated at 50,000 volts. The shortest wavelength limit of the X-rays produced is (a) \(0.1245 \AA\) (b) \(0.3485 \AA\) (c) \(0.2485 \AA\) (d) \(0.03456 \AA\)

Short answer

The shortest wavelength is option (c) 0.2485 Å.

Step-by-step solution

Understand the Relationship

The shortest wavelength (\(\lambda_{\text{min}}\)) of X-rays is given by the equation \(\lambda_{\text{min}} = \dfrac{hc}{eV}\), where \(h\) is Planck's constant, \(c\) is the speed of light, \(e\) is the electronic charge, and \(V\) is the voltage applied. Understanding this relationship is crucial to solving the problem.

Insert Known Values

Use the known values of the constants: \(h = 6.626 \times 10^{-34} \, \text{Js}\), \(c = 3 \times 10^8 \, \text{m/s}\), \(e = 1.602 \times 10^{-19} \, \text{C}\), and \(V = 50,000 \, \text{V}\). Substitute these values into the equation to find \(\lambda_{\text{min}}\).

Calculate the Wavelength

Calculate \(\lambda_{\text{min}}\) using the formula: \[\lambda_{\text{min}} = \dfrac{(6.626 \times 10^{-34})(3 \times 10^8)}{(1.602 \times 10^{-19})(50000)}\]. This simplifies to approximately:\[\lambda_{\text{min}} = 0.2485 \times 10^{-10} \, \text{m} = 0.2485 \, \text{Å}\].

Match with Given Options

Compare the calculated \(\lambda_{\text{min}}\) of \(0.2485 \, \text{Å}\) with the options provided. Option (c) matches our calculated shortest wavelength for the X-rays.

Key concepts

Planck's constant

Planck's constant is a fundamental constant in physics that links the energy of a photon to its frequency. It is denoted by the symbol \(h\) and has a value of \(6.626 \times 10^{-34} \, \text{Js}\) (joule seconds). This tiny number is essential in quantum mechanics and plays a big role in understanding how energy and frequency relate in wave mechanics.
For instance, when dealing with X-rays, Planck's constant helps to connect the energy change in an electron with the emission or absorption of photons of a certain frequency. The relationship is given by the formula \(E = h \times f\), where \(E\) is energy and \(f\) is frequency.

• This concept helps us understand various phenomena like the photoelectric effect, where electrons are emitted from a material when it absorbs light energy.
• Planck's constant is integral to the equation that determines the shortest wavelength limit of X-rays: \(\lambda_{\text{min}} = \dfrac{hc}{eV}\).

Speed of light

The speed of light, denoted as \(c\), is a universal constant important in many areas of physics, including optics and electromagnetism. It is the rate at which light waves propagate through the vacuum and is valued at \(3 \times 10^8 \, \text{m/s}\).
This constant is crucial when dealing with electromagnetic waves, including X-rays, as it helps determine their wavelengths and frequencies. When calculating the shortest wavelength limit in an X-ray, the speed of light is part of the equation \(\lambda_{\text{min}} = \dfrac{hc}{eV}\).

• The speed of light sets the maximum speed limit for all matter and information in the universe.
• It plays a key role in the theories of relativity, which explore the nature of space and time.

Electron volt

An electron volt (eV) is a unit of energy commonly used in the field of atomic and particle physics. It represents the amount of kinetic energy gained or lost by an electron when it moves across an electric potential difference of one volt.
With an approximate value of \(1.602 \times 10^{-19} \, \text{C}\) per electron volt, it provides a convenient way to measure very small amounts of energy on an atomic scale.

• In X-ray physics, the voltage applied to an X-ray tube is often measured in volts and directly affects the energy of the X-rays produced.
• When calculated in electron volts, this energy informs us about the wavelength of the emitted X-rays via the formula \(\lambda_{\text{min}} = \dfrac{hc}{eV}\), where \(V\) is the voltage in volts.

Shortest wavelength limit

The shortest wavelength limit is the smallest wavelength that X-rays can have when produced under certain conditions, such as a specific voltage applied in an X-ray tube. This is significant because it determines the energy and penetrating power of the X-rays.
The shortest wavelength limit can be calculated using the formula \(\lambda_{\text{min}} = \dfrac{hc}{eV}\). This equation outlines how the voltage, Planck's constant, and the speed of light come together to influence the wavelength. Given that higher voltages lead to shorter wavelengths, more energetic X-rays result.

• It is essential for applications in medical imaging and material analysis, where high energy X-rays are needed.
• By adjusting the voltage, one can control the wavelength and intensity of the X-rays for various diagnostic and research purposes.