Animated Solution for Physics - Dual Nature of Matter and Radiation: X-rays are produced in an X-ray tube operating at a given accelerating voltage. The wavelength of the continuous X-rays has values from

Imagine a beam of high-speed electrons hurtling towards a heavy metal target. As these electrons penetrate the target, they don't just crash into the atoms; they interact with the strong electric fields of the target nuclei. This interaction acts like a sudden brake, decelerating the electrons. According to classical electrodynamics, any accelerating or decelerating charge emits electromagnetic radiation. In this quantum realm, the lost kinetic energy is emitted as X-ray photons. This continuous spectrum of radiation is beautifully named Bremsstrahlung, which is German for "braking radiation."

Let's think about the extremes. What is the absolute maximum energy a single X-ray photon can carry away from this process? This extreme scenario occurs when an incoming electron is stopped completely in a single, head-on collision with a nucleus. All of its kinetic energy, which it gained from the accelerating voltage $V$, is transferred to a single photon.

The kinetic energy of the electron is $eV$. Therefore, the maximum photon energy is $E_{\text{max}}=eV$. Since the energy of a photon is inversely proportional to its wavelength ($E=\frac{hc}{\lambda }$), this maximum energy corresponds to the minimum possible wavelength, denoted as ${\lambda }_{\text{min}}$.

We can write this relationship as:

Because the accelerating voltage $V$ is a finite value, the minimum wavelength ${\lambda }_{\text{min}}$ must be strictly greater than zero (${\lambda }_{\text{min}}>0$). The spectrum has a sharp, hard cutoff at this exact value.

Now, let's flip the scenario. What happens if an electron only grazes a nucleus, experiencing a very weak interaction? It will lose only a tiny fraction of its kinetic energy. The emitted photon will carry away this infinitesimally small amount of energy.

As the energy of the emitted photon approaches zero ($E_{\text{min}}\to 0$), what happens to its wavelength? Again, using the inverse relationship $\lambda =\frac{hc}{E}$, as the denominator approaches zero, the wavelength stretches towards infinity (${\lambda }_{\text{max}}\to \infty$).

By analyzing the physical limits of the electron's energy loss, we've mapped out the entire continuous X-ray spectrum. The wavelength cannot be smaller than ${\lambda }_{\text{min}}$ (which is greater than zero), and it has no upper bound, extending all the way to infinity.

Therefore, the wavelength of continuous X-rays takes values from ${\lambda }_{\text{min}}$ to $\infty$, making option (b) the correct answer.