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?

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

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