The shortest wavelength of X-rays emitted from an X-ray tube depends on
(a) the current in the tube

Q: Why doesn't the shortest wavelength depend on the target material?

The shortest wavelength corresponds to the maximum energy a photon can have, which happens when an electron loses all its kinetic energy in a single collision. This kinetic energy is determined entirely by the accelerating voltage, not the target material. The target material affects the characteristic X-rays, not the continuous spectrum's cut-off.

Q: What happens if we increase the current in the tube?

Increasing the filament current increases the number of electrons emitted per second, which increases the intensity (number of photons) of the X-rays, but it does not change their maximum energy or shortest wavelength.

Q: What is the difference between continuous and characteristic X-rays?

Continuous X-rays (Bremsstrahlung) are produced by the deceleration of electrons and have a continuous range of wavelengths down to a cut-off. Characteristic X-rays are produced when target electrons are knocked out of inner shells, and their wavelengths depend on the target material.

Q: Why is it called the "cut-off" wavelength?

Because no X-ray photons can be produced with a wavelength shorter than this value. It represents the absolute maximum energy an electron can transfer to a single photon.

Q: Does the nature of the gas in the tube matter?

Modern X-ray tubes (Coolidge tubes) are highly evacuated (vacuum) so that electrons don't collide with gas molecules and lose energy before hitting the target. So, there is no gas to affect the process.

Question

The shortest wavelength of X-rays emitted from an X-ray tube depends on

Options:

(A) the current in the tube

(B) the voltage applied to the tube

(C) the nature of the gas in tube

(D) the atomic number of the target material

Sigma Percentile · Accepted Answer

### Analyzing the Setup
Imagine you are looking inside a Coolidge tube, the heart of an X-ray machine. Inside this highly evacuated glass tube, we have a cathode (a heated filament) and an anode (a heavy metal target). When we apply a high voltage $V$ across these electrodes, electrons are boiled off the cathode and violently accelerated towards the anode.

### The Master Equation
As an electron races towards the target, it gains kinetic energy. By the time it smashes into the anode, its maximum kinetic energy is given by:
$$K_{\max} = eV$$
where $e$ is the charge of the electron and $V$ is the accelerating voltage.

Now, what happens upon impact? The electron rapidly decelerates, and its kinetic energy is converted into electromagnetic radiation—X-ray photons. This process is known as *Bremsstrahlung* or braking radiation.

If an electron loses all of its kinetic energy in a single, head-on collision, it produces a single X-ray photon with the absolute maximum possible energy. 
$$E_{\max} = K_{\max} = eV$$

### Final Calculation
We know from the photon theory of light that the energy of a photon is inversely proportional to its wavelength:
$$E_{\max} = \frac{hc}{\lambda_{\min}}$$
where $h$ is Planck's constant, $c$ is the speed of light, and $\lambda_{\min}$ is the shortest (or cut-off) wavelength.

Equating the two expressions for maximum energy, we get:
$$\frac{hc}{\lambda_{\min}} = eV$$
$$\lambda_{\min} = \frac{hc}{eV}$$

Look closely at this final equation. The constants $h$, $c$, and $e$ never change. The only variable that dictates the shortest wavelength $\lambda_{\min}$ is the applied voltage $V$. It does not depend on the filament current, the nature of the gas (since it's a vacuum), or the atomic number of the target material.

Therefore, the shortest wavelength of X-rays emitted from an X-ray tube depends exclusively on the voltage applied to the tube.

Animated Solution for Physics - Dual Nature of Matter and Radiation: The shortest wavelength of X-rays emitted from an X-ray tube depends on

Select Answer:

Visualized Solution

$X$ -ray Tube Setup

Maximum Kinetic Energy

Shortest Wavelength (Cut-off)

Final Dependency

Conclusion

The Sigma Insight: Photon Theory of Light

Analyzing the Setup Imagine you are looking inside a Coolidge tube, the heart of an X-ray machine

The Master Equation As an electron races towards the target, it gains kinetic energy

Final Calculation

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