Animated Solution for Physics - Dual Nature of Matter and Radiation: When the number of electrons striking the anode of an X-ray tube is increased the ........ of the emitted X-rays increases, while when the speeds of the electrons striking the anode are increased the cut-off wavelength of the emitted X-rays ........ .

Imagine you are looking inside an X-ray tube. It is a fascinating piece of equipment where high-speed physics meets electromagnetic radiation. At one end, we have a cathode, which acts as an electron gun, firing a stream of electrons. At the other end is the anode, a heavy metal target.

When these electrons are accelerated across a high potential difference, they gain a tremendous amount of kinetic energy. The moment they slam into the anode, they undergo rapid deceleration. According to classical electrodynamics, accelerating or decelerating charges emit electromagnetic radiation. In this case, the deceleration is so sharp that the emitted radiation falls in the X-ray region of the electromagnetic spectrum.

The first part of our problem asks us to consider what happens when we increase the number of electrons striking the anode. Let's break this down logically.

Each electron that strikes the target has a certain probability of producing an X-ray photon. If we simply increase the number of electrons hitting the target every second—perhaps by increasing the filament current in the cathode—we are proportionally increasing the number of X-ray photons produced per second.

In physics, the total energy crossing a unit area per unit time is defined as intensity. Since the number of photons per second increases, the overall energy per second increases. Therefore, the intensity of the emitted X-rays increases. This gives us the answer to our first blank.

Now, let's shift our focus to the second part of the question: what happens when the speed of the electrons striking the anode is increased?

The speed of an electron is directly tied to its kinetic energy through the relation $K=\frac{1}{2}m{v}^2$. If the speed $v$ increases, the kinetic energy $K$ of the electron increases significantly.

When an electron strikes the target, it can lose its energy in multiple collisions, producing a continuous spectrum of X-rays. However, the most energetic X-ray photon is produced when an electron loses all of its kinetic energy in a single, head-on collision with a target atom.

By the principle of conservation of energy, the maximum energy of the emitted X-ray photon, $E_{\text{max}}$, is exactly equal to the kinetic energy $K$ of the incident electron. We know from Planck's quantum theory that the energy of a photon is inversely proportional to its wavelength:

Here, ${\lambda }_{\text{min}}$ is the shortest possible wavelength of the emitted X-rays, commonly referred to as the cut-off wavelength. Equating the two energies, we get:

Rearranging this to solve for the cut-off wavelength, we find:

This equation is our master key. It tells us that the cut-off wavelength is inversely proportional to the kinetic energy of the electrons. Since increasing the speed of the electrons increases their kinetic energy $K$, the denominator in our fraction grows larger. Consequently, the value of the fraction decreases.

Therefore, the cut-off wavelength of the emitted X-rays decreases. This perfectly fills our second blank, completing the conceptual puzzle!