Animated Solution for Physics - Atoms and Nuclei: Which one of the following statements is wrong in the context of X-rays generated from an X-ray tube ?

Continuous vs. Characteristic Spectra

Imagine a highly evacuated glass tube where electrons are boiled off a hot filament and accelerated across a massive potential difference, slamming into a heavy metal target. This violent collision produces X-rays, but the spectrum of these X-rays tells a fascinating two-part story.

As the high-speed electrons enter the target, they are strongly deflected and decelerated by the positively charged nuclei. According to classical electrodynamics, an accelerating (or decelerating) charge emits electromagnetic radiation. This "braking radiation" forms the continuous background of the X-ray spectrum.

The most energetic X-ray photon is produced when an electron loses all its kinetic energy in a single collision. The kinetic energy of the electron is given by $eV$, where $V$ is the accelerating voltage. Equating this to the energy of a photon gives us the Duane-Hunt law:

Rearranging for the cut-off wavelength, we get:

Notice something crucial here: ${\lambda }_{\text{min}}$ depends only on the accelerating voltage $V$. It has absolutely nothing to do with the atomic number $Z$ of the target material! This makes statement (b) in our problem fundamentally incorrect.

Superimposed on the continuous background are sharp, intense peaks. These are the characteristic X-rays. They occur when an incoming electron has enough energy to knock out an inner-shell electron (like from the K-shell) of a target atom.

When this happens, an electron from a higher energy shell (like the L or M shell) drops down to fill the vacancy. The energy difference between these two shells is emitted as an X-ray photon. Because these energy levels are unique to each element, the emitted X-rays are "characteristic" of the target material.

Moseley's Law beautifully describes this relationship:

Since frequency $ u$ is inversely proportional to wavelength $\lambda$, we can see that as the atomic number $Z$ increases, the energy of the emitted photon increases, and thus the wavelength $\lambda$ decreases. This confirms that statement (a) is perfectly correct.

Finally, what determines the overall brightness or intensity of the X-ray beam? Intensity is simply the total energy crossing a unit area per unit time, which depends on the number of X-ray photons emitted per second.

To get more photons, you need more electrons striking the target. You achieve this by increasing the current through the filament, which heats it up more and boils off more electrons via thermionic emission. This filament current is directly controlled by the electrical power supplied to the X-ray tube. Therefore, statement (c) is also correct.

By systematically breaking down the physics of the X-ray tube, we can confidently conclude that the cut-off wavelength of continuous X-rays is independent of the target's atomic number, making option (b) the wrong statement.