To tackle the problem regarding the cut-off wavelength of emitted rays in an X-ray tube when electrons with a de-Broglie wavelength strike a target, we first need to clarify some key concepts. Let's break this down step by step.

Understanding the Cut-off Wavelength

The cut-off wavelength, often denoted as λ_cut-off, is the maximum wavelength of X-rays produced when high-energy electrons collide with a target material. This wavelength corresponds to the minimum energy of the emitted X-rays. The relationship between energy and wavelength is given by the equation:

E = h * c / λ

Where:

• E is the energy of the photon,
• h is Planck's constant (approximately 6.626 x 10^-34 Js),
• c is the speed of light (approximately 3 x 10⁸ m/s),
• λ is the wavelength.

For X-ray production, the maximum energy of the emitted X-rays corresponds to the kinetic energy of the incoming electrons, which can be expressed as:

E = e * V

Where:

• e is the charge of the electron (approximately 1.602 x 10^-19 C),
• V is the accelerating voltage in volts.

Calculating the Cut-off Wavelength

By equating the two expressions for energy, we can derive the cut-off wavelength:

e * V = h * c / λ_cut-off

Rearranging this gives us:

λ_cut-off = h * c / (e * V)

This equation allows us to calculate the cut-off wavelength if we know the accelerating voltage of the X-ray tube.

Defining Key Terms

1. Cut-off Wavelength

The cut-off wavelength is the longest wavelength of X-rays that can be emitted from an X-ray tube. It represents the threshold below which all emitted X-rays have higher energy. This is significant because it indicates the maximum energy that can be produced by the electrons striking the target. Any X-ray emitted with a wavelength longer than this cut-off does not have enough energy to be produced under the given conditions.

2. Characteristic X-rays

Characteristic X-rays are specific wavelengths of X-rays emitted by a material when its inner-shell electrons are ejected, and outer-shell electrons transition to fill these vacancies. This process occurs when high-energy electrons collide with the target material, causing ionization. The energy difference between the shells corresponds to the energy of the emitted X-ray, resulting in discrete wavelengths characteristic of the target element. For example, if you strike a copper target, the emitted X-rays will have specific wavelengths that are unique to copper.

3. Momentum of a Photon

The momentum of a photon can be calculated using the formula:

p = E / c

Where:

• p is the momentum,
• E is the energy of the photon,
• c is the speed of light.

Since the energy of a photon can also be expressed in terms of its wavelength (E = h * c / λ), we can rewrite the momentum in terms of wavelength:

p = h / λ

This relationship shows that the momentum of a photon is inversely proportional to its wavelength. Thus, shorter wavelengths (higher energy) correspond to greater momentum.

In summary, understanding the cut-off wavelength and the concepts of characteristic X-rays and photon momentum is crucial for grasping the principles behind X-ray production and the behavior of electromagnetic radiation. If you have any further questions or need clarification on any of these topics, feel free to ask!