Hard X-rays for the study of fractures in bon | vedclass

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Question

(B) The energy of an $X$-ray photon is given by $E = \frac{hc}{\lambda}$.For the minimum wavelength $\lambda_{min}$,the accelerating voltage $V$ is gi

Vedclass · Accepted Answer

$> 124.2 \, kV$

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(B) The energy of an $X$-ray photon is given by $E = \frac{hc}{\lambda}$.For the minimum wavelength $\lambda_{min}$,the accelerating voltage $V$ is given by $eV = \frac{hc}{\lambda_{min}}$.Substituting the values $h = 6.63 	imes 10^{-34} \, J \cdot s$,$c = 3 	imes 10^8 \, m/s$,$e = 1.6 	imes 10^{-19} \, C$,and $\lambda = 10^{-11} \, m$:$V = \frac{6.63 	imes 10^{-34} 	imes 3 	imes 10^8}{1.6 	imes 10^{-19} 	imes 10^{-11}}$$V = \frac{19.89 	imes 10^{-26}}{1.6 	imes 10^{-30}} = 12.43 	imes 10^4 \, V = 124.3 \, kV$.Since the wavelength must be at least $10^{-11} \, m$,the energy of the electrons must be sufficient to produce this wavelength. Therefore,the accelerating voltage must be at least $124.3 \, kV$ to produce wavelengths equal to or shorter than this limit. However,in the context of the question asking for the condition to produce this specific minimum wavelength,the voltage must be at least $124.2 \, kV$ (using $hc \approx 1242 \, eV \cdot nm$). Thus,$V \geq 124.2 \, kV$.

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 the $X$-ray machine should be:

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