Animated Solution for Physics - Atoms and Nuclei: The intensity of X-rays from a coolidge tube is plotted against wavelength $\lambda$ as shown in the figure. The minimum wavelength found is ${\lambda }_c$ and the wavelength of the $K_{\alpha }$ line is ${\lambda }_k$. As the accelerating voltage is increased

Imagine you are standing inside a Coolidge tube, watching high-energy electrons smash into a heavy metal target. The graph we see in this problem is a beautiful combination of two entirely different physical phenomena happening simultaneously.

First, we have the continuous spectrum. As the incoming electrons get deflected and decelerated by the strong electric fields of the target nuclei, they lose kinetic energy and emit photons. This is called Bremsstrahlung or braking radiation. The absolute maximum energy a photon can have occurs when an electron loses all its kinetic energy in a single collision. This maximum energy corresponds to the minimum wavelength, known as the cut-off wavelength, ${\lambda }_c$.

Second, we have the characteristic peaks. Sometimes, an incoming electron has enough energy to knock out an inner-shell electron from a target atom. When a higher-shell electron drops down to fill this vacancy, it emits an X-ray photon with a very specific, quantized energy. These are the sharp spikes on the graph, like the $K_{\alpha }$ line at ${\lambda }_k$. These peaks are the unique fingerprints of the target atoms.

To understand how the graph shifts, we need to look at the mathematical engines driving these two wavelengths.

The cut-off wavelength depends entirely on the accelerating voltage $V$ that gave the incoming electrons their kinetic energy in the first place:

On the other hand, the characteristic wavelength ${\lambda }_k$ is governed by Moseley's Law and the Bohr model transitions. It depends strictly on the atomic number $Z$ of the target material:

The problem states that we are increasing the accelerating voltage $V$. Let's analyze the impact on our two key players.

Since we are only changing the voltage and not swapping out the metal target, the atomic number $Z$ remains perfectly constant. Because $Z$ is constant, the energy levels of the target atoms don't change. Therefore, the characteristic wavelength ${\lambda }_k$ remains completely unchanged. It is anchored in place.

However, look at the formula for ${\lambda }_c$. The voltage $V$ sits squarely in the denominator. As $V$ increases, the fraction $\frac{hc}{eV}$ must decrease. Therefore, ${\lambda }_c$ decreases, shifting the starting point of the continuous spectrum to the left.

We are asked to evaluate what happens to the difference ${\lambda }_k-{\lambda }_c$.

We have established that ${\lambda }_k$ is a constant value, and ${\lambda }_c$ is becoming a smaller value. If you subtract a smaller number from a constant number, the result must grow larger.

Mathematically:

Thus, the gap between the characteristic peak and the cut-off wavelength widens. The correct answer is that ${\lambda }_k-{\lambda }_c$ increases.