A-level Physics (Advancing Physics)/Electron Behaviour as a Quantum Phenomenon/Worked Solutions

1. An electron moves at 30,000 ms⁻¹. What is its de Broglie wavelength?

$\lambda =\frac{h}{mv}=\frac{6.626\times 10^{-34}}{9.1\times 10^{-31}\times 30,000}=2.43\times 10^{-8}\mathrm{m}$

2. What is its frequency?

$f=\frac{E_{\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{c}}}{h}=\frac{\frac{1}{2}9.1\times 10^{-31}\times 30,000^2}{6.626\times 10^{-34}}=6.18\times 10^{11}\mathrm{H}\mathrm{z}$

3. What is its kinetic energy, in eV?

From the top half of the fraction in the previous question:

$E_{\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{c}}=\frac{1}{2}9.1\times 10^{-31}\times 30,000^2=4.10\times 10^{-22}\mathrm{J}=\frac{4.10\times 10^{-22}}{1.6\times 10^{-19}}\mathrm{e}\mathrm{V}=2.56\mathrm{m}\mathrm{e}\mathrm{V}$

4. Given that it is travelling out of an electron gun, what was the potential difference between the anode and the cathode?

2.56 mV – that's why we use eV!

5. An electron is accelerated by a potential difference of 150 V. What is its frequency?

$E_{\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{c}}=1.6\times 10^{-19}\times 150=2.4\times 10^{-17}\mathrm{J}$

$f=\frac{E_{\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{c}}}{h}=\frac{2.4\times 10^{-17}}{6.626\times 10^{-34}}\mathrm{H}\mathrm{z}=3.62\times 10^{16}\mathrm{H}\mathrm{z}$