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?

$f={\frac {E_{\mathrm {kinetic} }}{h}}={\frac {{\tfrac {1}{2}}\,9.1\times 10^{-31}\times 30{,}000^{2}}{6.626\times 10^{-34}}}=6.18\times 10^{11}\,\mathrm {Hz}$

3. What is its kinetic energy, in eV?

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

$E_{\mathrm {kinetic} }={\tfrac {1}{2}}\,9.1\times 10^{-31}\times 30{,}000^{2}=4.10\times 10^{-22}\,\mathrm {J} ={\frac {4.10\times 10^{-22}}{1.6\times 10^{-19}}}\,\mathrm {eV} =2.56\,\mathrm {meV}$

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?

$E_{\mathrm {kinetic} }=1.6\times 10^{-19}\times 150=2.4\times 10^{-17}\,\mathrm {J}$

$f={\frac {E_{\mathrm {kinetic} }}{h}}={\frac {2.4\times 10^{-17}}{6.626\times 10^{-34}}}\,\mathrm {Hz} =3.62\times 10^{16}\,\mathrm {Hz}$