If the potential difference applied to an X-ray tube is doubled while

If the potential difference applied to an X-ray tube is doubled while keeping the separation between the filament and the target as same, what will happen to the cutoff wavelength?

Will remain same

Will be doubled

Will be halved

Will be four times of the original wavelength

Answer is Right!

Answer is Wrong!

This question was previously asked in

UPSC NDA-1 – 2017

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The question asks what happens to the cutoff wavelength of X-rays produced in an X-ray tube if the potential difference is doubled while keeping the filament-target separation the same.

The production of X-rays in an X-ray tube involves accelerating electrons through a potential difference ($V$) and making them strike a target. The energy gained by an electron accelerated through potential $V$ is $eV$, where $e$ is the elementary charge. This energy is converted into electromagnetic radiation (X-rays) and heat. The minimum wavelength ($\lambda_{min}$) of the emitted X-rays, also known as the cutoff wavelength or Duane-Hunt limit, occurs when the entire energy of the electron is converted into a single X-ray photon. The energy of a photon is $hf = hc/\lambda$, where $h$ is Planck’s constant, $f$ is frequency, $c$ is the speed of light, and $\lambda$ is the wavelength.
According to the Duane-Hunt law: $eV = hc/\lambda_{min}$
This equation shows that the cutoff wavelength $\lambda_{min}$ is inversely proportional to the applied potential difference $V$: $\lambda_{min} \propto 1/V$.
If the potential difference is doubled, i.e., $V’ = 2V$, the new cutoff wavelength $\lambda’_{min}$ will be:
$\lambda’_{min} = \frac{hc}{e(2V)} = \frac{1}{2} \left(\frac{hc}{eV}\right) = \frac{1}{2} \lambda_{min}$
So, the cutoff wavelength will be halved. The separation between the filament and the target does not directly affect the cutoff wavelength, which is determined by the maximum energy of the electrons reaching the target, dictated by the potential difference.

The X-ray spectrum produced consists of a continuous spectrum (bremsstrahlung) with a minimum wavelength $\lambda_{min}$ and characteristic peaks at specific wavelengths. The cutoff wavelength depends only on the accelerating voltage, while the intensity and characteristic peaks also depend on the target material and the electron beam current.

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