Solveeit Logo
Login

Question

Question: An X-ray tube emits X-rays of wavelength \[0.1\,{A^0}\] under the application of a potential differe...

An X-ray tube emits X-rays of wavelength 0.1 A00.1\,{A^0} under the application of a potential difference of
A. 12.4 kV
B. 24.8 kV
C. 124 kV
D. 200 kV

Explanation

Solution

Calculate the energy of the X-ray radiation using the formula for energy of the electromagnetic radiation in terms of wavelength of the wave. When the electron moves across the potential difference in the X-ray tube, the energy obtained by the electron is eΔVe\Delta V. This is the same energy required to emit the X-ray radiation in the tube.

Formula used:
The energy E of electromagnetic radiation of wavelength λ\lambda is given as,
E=hcλE = \dfrac{{hc}}{\lambda }
Here, h is Planck’s constant and c is the speed of light.

Complete step by step answer:
We know that energy E of electromagnetic radiation of wavelength λ\lambda is given as,
E=hcλE = \dfrac{{hc}}{\lambda }
Here, h is Planck’s constant and c is the speed of light.
Therefore, we can calculate the energy of the X-ray radiation by substituting 1240 eV nm1240\,eV\,nm for hchc and 0.01 nm for λ\lambda in the above equation.
E=1240 eV nm0.01 nmE = \dfrac{{1240\,eV\,nm}}{{0.01\,nm}}
⇒E=124000 eV\Rightarrow E = 124000\,eV
We know that the energy obtained by the electron in the X-ray tube across the potential difference ΔV\Delta V is,
E=eΔVE = e\Delta V
⇒ΔV=Ee\Rightarrow \Delta V = \dfrac{E}{e}
Here, e is the charge on an electron.
This same amount of energy is possessed by X-ray radiation.
We can substitute E=124000 eVE = 124000\,eV in the above equation.
ΔV=124000 eVe\Delta V = \dfrac{{124000\,eV}}{e}
⇒ΔV=124000 V\Rightarrow \Delta V = 124000\,V
∴ΔV=124kV\therefore \Delta V = 124kV

So, the correct answer is “Option C”.

Additional Information:
In X-ray tubes, when an electron travels in the potential difference of ΔV\Delta V from one end to another, the energy obtained by the electron is eΔVe\Delta V. When this electron strikes the end of the tube, it gives all of its energy to emit X-ray radiation. The energy radiated by the X-ray is the same as energy obtained by the electron in potential difference.

Note:
If you want to calculate the energy of x-ray radiation in joule using the formulaE=hcλE = \dfrac{{hc}}{\lambda }, you can substitute the values of Planck’s constant and speed of light separately. Students should remember the value of 1 A0=10−10 m1\,{A^0} = {10^{ - 10}}\,m. To use the above formula, the wavelength should be in nm if you want to calculate the energy in eV.