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Dual Nature of Radiation and Matter — Physics STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 SciencePhysicsDual Nature of Radiation and Matter3 Marks
Question
An electron microscope uses electrons accelerated by a voltage of 50 kV. Determine the de Broglie wavelength associated with the electrons. If other factors (such as numerical aperture, etc.) are taken to be roughly the same, how does the resolving power of an electron microscope compare with that of an optical microscope which uses yellow light?
✓

Answer

Given,
Accelerating voltage, $V = 50\ kV$
$\therefore$ Energy of electrons, $E = eV = 50\ keV$
or, Energy, $E = 50 \times 10^3 \times 1.6 \times 10^{-19} J$
De-broglie wavelength is given by, $\lambda=\frac{\text{h}}{\sqrt{2\text{mE}}}$
$\Rightarrow\ \lambda=\frac{6.62\times10^{-34}}{\sqrt{2\times9.1\times10^{-31}\times50\times10^{3}\times1.6\times10^{-19}}}$
$\lambda=\frac{6.62\times10^{-34}}{1.21\times10^{-22}}=5.47\times10^{-12}\ \text{m}.$
Resolving power of microscope $\propto\ \frac{1}{\lambda}$
$\therefore\ \frac{\text{R.P. of electron microscope}}{\text{R.P. of optical microscope}}=\frac{\lambda\text{y}}{\lambda}+\frac{5.9\times10^{-7}}{5.47\times10{-17}}\cong10^{+5}$
where, $\lambda\text{y}$ = wavelength yellow light.
Resolving power of a microscope is inversely proportional to the wavelength of the radiation used.
Since the wavelength of yellow light is $5.99 \times 10^{-7} m$, so it follows that electron microscope will have resolving power $10^5$ times that of optical microscope.

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