Animated Solution for Physics - Atoms and Nuclei: Assertion If the accelerating potential in an X-ray tube is increased, the wavelengths of the characteristic X-rays do not change. Reason When an electron beam strikes the target in an X-ray tube, part of the kinetic energy is converted into X-ray energy.

When a high-energy beam of electrons strikes a heavy metal target in an X-ray tube, a fascinating spectrum of electromagnetic radiation is emitted. If we plot the intensity of these X-rays against their wavelength, we see two very distinct features: a smooth, continuous background curve and sharp, towering spikes.

The continuous background is known as Bremsstrahlung, or "braking radiation." As the fast-moving electrons penetrate the target, they are deflected and decelerated by the strong electric fields of the target nuclei. According to classical electrodynamics, an accelerating (or decelerating) charge emits radiation. Because the electrons undergo varying degrees of deceleration, they emit photons of varying energies, creating a continuous spectrum.

The most energetic photon is produced when an electron loses all its kinetic energy in a single collision. This maximum energy corresponds to a minimum wavelength, known as the cut-off wavelength (${\lambda }_{\text{min}}$). Mathematically, this is given by:

where $V$ is the accelerating potential. Notice that ${\lambda }_{\text{min}}$ is inversely proportional to $V$. If we increase the accelerating potential, the electrons strike the target with more kinetic energy, and the cut-off wavelength shifts to a smaller value.

Superimposed on this continuous background are the sharp spikes known as characteristic X-rays. These are produced by a completely different mechanism. When a highly energetic incident electron collides with an inner-shell electron (like a K-shell electron) of a target atom, it can knock it completely out of the atom. This leaves a vacancy in the inner shell.

To restore stability, an electron from a higher energy level (like the L or M shell) drops down to fill the vacancy. The energy difference between these two discrete atomic levels is emitted as an X-ray photon:

Because the energy levels of an atom are quantized and unique to each element (depending on its atomic number $Z$), the wavelengths of these emitted photons are strictly characteristic of the target material. They do not depend on the accelerating potential, provided the potential is high enough to knock out the inner electron in the first place.

Let's look at the Assertion: If the accelerating potential in an X-ray tube is increased, the wavelengths of the characteristic X-rays do not change.

As we just established, the characteristic wavelengths are locked to the atomic energy levels of the target material. Increasing the accelerating potential changes the continuous spectrum's cut-off wavelength, but the characteristic peaks remain exactly where they are. Thus, the Assertion is True.

Now, let's look at the Reason: When an electron beam strikes the target in an X-ray tube, part of the kinetic energy is converted into X-ray energy.

This is a fundamental fact. The kinetic energy of the striking electrons is indeed converted into X-ray photons (both continuous and characteristic) and a massive amount of heat. Thus, the Reason is also True.

However, does the Reason explain the Assertion? No. The Reason merely states the general principle of energy conservation in an X-ray tube. It does not explain why the characteristic wavelengths are immune to changes in the accelerating potential. The true explanation lies in the quantized nature of atomic energy levels.

Therefore, both the Assertion and the Reason are true, but the Reason is not the correct explanation for the Assertion.