Question

Through what potential difference must electrons be accelerated so that they will exhibit wave nature in passing through a pinhole 0.10 micrometers in diameter?​

Answer 1

Answer:

0.15 mV

Explanation:

In order to exhibit wave nature, the de Broglie wavelength of the electron must be of the same size of the diameter of the pinhole, therefore:

$\lambda=0.10 \mu m = 1.0\cdot 10^{-7} m$

The de Broglie wavelength of an electron is

$\lambda = \frac{h}{mv}$

where

$h=6.63\cdot 10^{-34} Js$ is the Planck constant

$m=9.11\cdot 10^{-31} kg$ is the mass of the electron

v is the electron's speed

Therefore, the electron's speed must be

$v=\frac{h}{m\lambda}=\frac{6.63\cdot 10^{-34}}{(9.11\cdot 10^{-31})(1.0\cdot 10^{-7})}=7278 m/s$

When accelerated through a potential difference $\Delta V$, the kinetic energy gained by the electron is equal to the change in electric potential energy, therefore

$e\Delta V = \frac{1}{2}mv^2$

where

$e=1.6\cdot 10^{-19}$ is the magnitude of the charge of the electron

So, we can find the potential difference needed:

$\Delta V=\frac{mv^2}{2e}=\frac{(9.11\cdot 10^{-31})(7278)^2}{2(1.6\cdot 10^{-19})}=1.5\cdot 10^{-4}V = 0.15 mV$