Question
Find the,
- maximum frequency, and
- minimum wavelength of $X-$ rays produced by $30 kV$ electrons.
✓
Answer
Potential of the electrons $, V = 30 kV = 3 \times 10^4 V$
Hence, energy of the electrons $, E = 3 \times 10^4 eV$
Where $, e = $ Charge on an electron $= 1.6 \times 10^{-19} C$
$E = hv$
Where,
$h =$ Planck's constant $= 6.626 \times 10-34 Js$
$\therefore\ \text{v}=\frac{\text{E}}{\text{h}}$
$=\frac{1.6\times10^{-19}\times3\times10^4}{6.626\times10^{-34}}=7.24\times10^{18}\ \text{Hz}$
Hence, the maximum frequency of $X-$ rays produced is $ 7.24 \times 10^{18} Hz$.
$=\frac{3\times10^{8}}{7.24\times10^{18}}=4.14\times10^{-11}\ \text{m}=0.0414\ \text{nm}$
Hence, the minimum wavelength of $X-$ rays produced is $ 0.0414\ nm$.
Hence, energy of the electrons $, E = 3 \times 10^4 eV$
Where $, e = $ Charge on an electron $= 1.6 \times 10^{-19} C$
- Maximum frequency produced by the $X-$ rays $= v$
$E = hv$
Where,
$h =$ Planck's constant $= 6.626 \times 10-34 Js$
$\therefore\ \text{v}=\frac{\text{E}}{\text{h}}$
$=\frac{1.6\times10^{-19}\times3\times10^4}{6.626\times10^{-34}}=7.24\times10^{18}\ \text{Hz}$
Hence, the maximum frequency of $X-$ rays produced is $ 7.24 \times 10^{18} Hz$.
- The minimum wavelength produced by the $X-$ rays is given as:
$=\frac{3\times10^{8}}{7.24\times10^{18}}=4.14\times10^{-11}\ \text{m}=0.0414\ \text{nm}$
Hence, the minimum wavelength of $X-$ rays produced is $ 0.0414\ nm$.
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