Question

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

Answer 1

Hint: 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 \[e\Delta V\]. 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 \[\lambda \] is given as,
\[E = \dfrac{{hc}}{\lambda }\]
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 \[\lambda \] is given as,
\[E = \dfrac{{hc}}{\lambda }\]
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 \[1240\,eV\,nm\] for \[hc\] and 0.01 nm for \[\lambda \] in the above equation.
\[E = \dfrac{{1240\,eV\,nm}}{{0.01\,nm}}\]
\[ \Rightarrow E = 124000\,eV\]
We know that the energy obtained by the electron in the X-ray tube across the potential difference \[\Delta V\] is,
\[E = e\Delta V\]
\[ \Rightarrow \Delta V = \dfrac{E}{e}\]
Here, e is the charge on an electron.
This same amount of energy is possessed by X-ray radiation.
We can substitute \[E = 124000\,eV\] in the above equation.
\[\Delta V = \dfrac{{124000\,eV}}{e}\]
\[ \Rightarrow \Delta V = 124000\,V\]
\[ \therefore \Delta V = 124kV\]

So, the correct answer is “Option C”.

Additional Information:
In X-ray tubes, when an electron travels in the potential difference of \[\Delta V\] from one end to another, the energy obtained by the electron is \[e\Delta V\]. 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 formula\[E = \dfrac{{hc}}{\lambda }\], you can substitute the values of Planck’s constant and speed of light separately. Students should remember the value of \[1\,{A^0} = {10^{ - 10}}\,m\]. To use the above formula, the wavelength should be in nm if you want to calculate the energy in eV.