Question

Hard X-rays for the study of fractures in bones should have a minimum wavelength of ${{10}^{-11}}m$. The accelerating voltage for electrons in X-rays machine should be
A. <124.2 kV
B. >124.2 kV
C. between 60 kV and 70 kV
D. =100 kV

Answer 1

Hint: Use the formula for the maximum energy of an X-rays i.e $h\nu =eV$. Use the formula $\nu =\dfrac{c}{\lambda }$ and substitute for frequency. If the wavelength is minimum then the accelerating voltage is maximum.

Formula used:
$h\nu =eV$
$\nu =\dfrac{c}{\lambda }$

Complete answer:
X-rays belong to the electro-magnetic spectrum of the rays emitted by the sun. Other than receiving X-rays from the sun, we can also produce the X-rats in the labs.
We know that electromagnetic waves consist of massless particles called photons. When electrons accelerate, they emit the X-rays in the form of photons. Each electron emits one photon.
Now, the electrons are accelerated with the help of a potential difference called as accelerating voltage.
We know that a photon has an energy of $E=h\nu $, where h is Planck's constant and $\nu $ is the frequency. Therefore, an X-ray has an energy of $E=h\nu $. And this energy is given by the accelerating voltage.
Therefore,
$h\nu =eV$…. (i), where the charge on an electron and V is the accelerating voltage.
We generally categorise the X-rays into hard X-rays and soft X-rays. Hard X-rays are those X-rays that have the maximum energy. Soft X-rays are those X-rays that have minimum energy.
If hard X-rays have maximum energy means that they have maximum frequency. And for maximum energy the accelerating voltage must be maximum.
The relation between frequency and wavelength ($\lambda $) is given as $\nu =\dfrac{c}{\lambda }$, where c is the speed of light in vacuum and it is constant.
Therefore, wavelength of the X-rays is inversely proportional to the frequency of the X-rays. This means that if the frequency of the rays is maximum then its wavelength is minimum.
Substitute the value of frequency in equation (i).
 $h\dfrac{c}{{{\lambda }_{\min }}}=e{{V}_{\max }}$ …. (ii).
The values of h, c and e are $6.625\times {{10}^{-34}}Js$, $3\times {{10}^{8}}m{{s}^{-1}}$ and $1.6\times {{10}^{-19}}C$ respectively. The given value of ${{\lambda }_{\min }}={{10}^{-11}}m$.
Substitute the values in equation (ii).
$\Rightarrow \left( 6.625\times {{10}^{-34}} \right)\dfrac{3\times {{10}^{8}}}{{{10}^{-11}}}=\left( 1.6\times {{10}^{-19}} \right){{V}_{\max }}$
$\Rightarrow {{V}_{\max }}=\left( 6.625\times {{10}^{-34}} \right)\dfrac{3\times {{10}^{8}}}{\left( 1.6\times {{10}^{-19}} \right){{10}^{-11}}}=124000V=124kV$
This is the maximum voltage that can be applied. Therefore, the accelerating voltage must be less than 124kV.

So, the correct answer is “Option A”.

Note:
Note may make a mistake in writing the correct option even after finding the correct value of maximum voltage.
Note the value that we found the value of the maximum voltage. This means that this is the maximum value of the voltage to get the X-rays of minimum wavelength. If the voltage is greater than the maximum voltage then the wavelength will be less than the minimum. Hence, the value of the accelerating voltage must be less than 124kV.