Question

A) Find the maximum frequency of X-rays produced by $30kV$ electrons.

B) Find the minimum wavelength of X-rays produced by $30kV$ electrons.

Answer 1

A) Given:
Potential of the electrons,
$V=30kV=3\times {10}^{4}V$
Hence, energy of the electrons,
$E=3\times {10}^{4}eV$,
As we know,
$1eV=1.6\times {10}^{-19}J$
Therefore,
$E=3\times {10}^4\times 1.6\times {10}^{-19}J$
$E=4.8\times {10}^{-15}J$

Then the energy of the electrons is given by the relation,
$E=hv$
$v=\frac{E}{h}$
Where $h$ = Planck's constant
$h=6.626\times {10}^{-34}Js$
Hence, the maximum frequency of X - rays produced is,
$v_{max}=\frac{4.8\times {10}^{-15}}{6.626\times {10}^{-34}}$
$v_{max}=7.24\times {10}^{18}Hz$
Final Answer : $7.24\times {10}^{18}Hz$.

B) The minimum wavelength produced by the X-rays is given by,
${\lambda }_{min}=\frac{c}{v_{max}}$
${\lambda }_{min}=\frac{3\times {10}^8}{7.24\times {10}^{18}}$
${\lambda }_{min}=4.14\times {10}^{-11}m$
${\lambda }_{min}=0.0414nm$
Hence, the minimum wavelength of X - rays produced is 0.0414 nm.
Final Answer: ${\lambda }_{min}=0.0414nm$.