De Broglie Wavelength And Accelerating Voltage

When an electron is accelerated from rest through a potential difference $V$, NCERT shows that its de Broglie wavelength simplifies to the handy formula $\lambda =\frac{1.227}{\sqrt{V}}$ nm, obtained by equating the gained kinetic energy $eV$ to $\frac{p^2}{2m}$ and substituting into $\lambda =\frac{h}{p}$. For $V=100$ V, $\sqrt{V}=10$, so $\lambda =\frac{1.227}{10}=0.1227\approx 0.123$ nm. This sub-nanometre wavelength is why accelerated electrons are ideal for probing crystal structures. The value 1.23 nm is wrong because it forgets to take the square root of the voltage, leaving the wavelength ten times too large. The value 0.0123 nm is wrong because it divides by 100 instead of by $\sqrt{100}=10$. The value 12.3 nm is wrong because it both omits the square root and misplaces a decimal, giving a wildly large figure. As stated in NCERT Class 12, Chapter 11, the de Broglie wavelength of accelerated electrons decreases as the accelerating voltage increases. This compact formula is derived by combining the kinetic-energy relation with the de Broglie definition, and it is valid only in the non-relativistic regime where the accelerating voltage is modest. It underpins the electron microscope, whose resolving power far exceeds that of an optical microscope precisely because these accelerated electrons have sub-nanometre wavelengths. A magnitude check confirms that a 100 V electron has a wavelength of about 0.12 nm, close to atomic spacing and well suited to electron diffraction.