Problem 84 - Master Resource Book in JEE Main Physics

Question

Which one of the following statements is wrong in the context of X-rays generated from a X-rays tube? (a) Wavelength of characteristic X-rays decreases when the atomic number of the target increases (b) Cut-off wavelength of the continuous \(X\)-rays depends on the atomic number of the target (c) Intentsity of the characteristic \(X\)-rays depends on the electrical power given to the \(X\)-rays tube (d) Cut-off wavelength of the continuous X-rays depends on the energy of the electrons in the \(X\)-ray tube

Step-by-Step Solution

Answer

Statement (b) is wrong as the cut-off wavelength does not depend on the target's atomic number.

Step 1: Understanding Cut-off Wavelength

The cut-off wavelength (
f3_{min}
f3) for X-rays is inversely related to the maximum energy (
f3) of the electrons striking the target in an X-ray tube. It is unrelated to the atomic number of the target. Therefore, statement (b) is potentially incorrect because the cut-off wavelength does not depend on the atomic number but rather on the energy of the electrons.

Step 2: Analyzing Characteristic X-rays

Characteristic X-rays are derived from specific differences in energy levels within an atom and are dependent on the atomic number (Z) of the target. An increase in the atomic number of the target results in a greater energy difference between inner electron shells, thus decreasing the wavelength of the emitted X-rays. Therefore, statement (a) is correct.

Step 3: Intensity and Electrical Power

The intensity of characteristic X-rays can increase with the electrical power supplied to the X-ray tube. Higher power results in more electrons striking the target, which increases the likelihood of generating characteristic X-rays. Hence, statement (c) is correct.

Step 4: Evaluating Energy Dependence

The cut-off wavelength of continuous X-rays only depends on the energy of the electrons, as expressed by
f 3_{min} =
f 3_{c}
f 3/hc, where
f 3 is the electron charge,
f 3 is the applied voltage and hc is a constant. Thus, statement (d) is correct, meaning it does depend on electron energy.

Key Concepts

Characteristic X-rays

Characteristic X-rays are unique to the element from which they are emitted. When fast-moving electrons hit the target in an X-ray tube, they can knock out electrons from the inner shells of target atoms. When electrons from outer shells drop into the inner shells to fill the vacancy, X-rays are emitted. These X-rays are called characteristic because their energy, and thus their wavelength, is specific to the differences in energy levels of the atomic shells involved. - The wavelength of characteristic X-rays is inversely related to the atomic number ( - Larger atomic numbers lead to larger energy differences between shells, resulting in shorter wavelengths of emitted X-rays.

Cut-off Wavelength

In the context of X-ray production in a tube, the cut-off wavelength refers to the shortest possible wavelength (or maximum energy) of the X-rays that can be produced. This cut-off is determined solely by the energy of the electrons that are accelerated in the X-ray tube and strike the target. - The energy of an electron depends on the voltage applied across the X-ray tube. - As a result, the cut-off wavelength, \( \lambda_{\text{min}} \), is described by the equation \( E = \frac{hc}{\lambda_{\text{min}}} \), where \( E \) is the electron energy determined by voltage, \( h \) is Planck's constant, and \( c \) is the speed of light. - This explains why the cut-off wavelength does not depend on the atomic number of the target but solely on the applied voltage and electron energy.

Atomic Number Effects

The atomic number ( - A higher atomic number causes greater binding energy differences between the inner electron shells. - Consequently, the characteristic X-ray wavelengths are shorter as energy differences increase with higher atomic numbers. - This changing wavelength with atomic number allows scientists to identify elements through X-ray spectroscopy, utilizing their unique characteristic X-rays.

X-ray Tube Operation

Understanding the operation of an X-ray tube is crucial for comprehending the emission of X-rays. In an X-ray tube, electrons are emitted from a heated filament and are accelerated toward a metal target, usually tungsten, by a high voltage. Upon striking the target, these high-energy electrons can produce X-rays through two primary processes: - Characteristic X-rays result from the reorganization of electrons within the target atoms, specific to each element. - Continuous X-rays, or bremsstrahlung, result from the deceleration of electrons in the target material, producing a range of X-ray wavelengths. The efficiency and intensity of X-ray production depend on: - The voltage applied, which determines the maximum energy and, consequently, the cut-off wavelength. - The current flowing through the tube, which influences the number of electrons hitting the target. - The target material, affecting both the characteristic X-ray lines and overall X-ray output efficiency.