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

1. How much energy does a photon with a frequency of 50 kHz carry?

$$E = hf = 6.626 \times 50 \times 10^3 \times 10^{-34}= 3.313 \times 10^{-29}\mbox{J}\,$$

'''2. A photon carries 10−30J of energy. What is its frequency?'''

$$f = \frac{E}{h} = \frac{10^{-30}}{6.626 \times 10^{-34}} \approx 1509\mbox{Hz}$$

3. How many photons of frequency 545 THz does a 20W monochromatic bulb give out each second?

First calculate the amount of energy given out per. second:

$$P = \frac{\Delta E}{t}$$

$$\Delta E = Pt = 20 \times 1 = 20\mbox{J}\,$$

Then, calculate the amount of energy carried by each photon:

$$E = hf = 6.626 \times 545 \times 10^{12} \times 10^{-34} \approx 3.61 \times 10^{-19}\mbox{J}\,$$

Then divide the former by the latter to give the number of photons n:

$$n = \frac{20}{3.61 \times 10^{-19}} \approx 5.54 \times 10^{19}\mbox{ photons}$$

'''4. In one minute, a monochromatic bulb gives out a million photons of frequency 600 THz. What is the power of the bulb?'''

First calculate the energy carried by one photon:

$$E = hf = 6.626 \times 10^{-34} \times 600 \times 10^{12} \approx 3.98 \times 10^{-19}\mbox{J}\,$$

Then work out the energy carried by 1,000,000 photons:

$$E = 10^6 \times 3.98 \times 10^{-19} = 3.98 \times 10^{-13}\mbox{J}$$

Then work out the power of the bulb:

$$P = \frac{\Delta E}{t} = \frac{3.98 \times 10^{-13}}{60} = 6.63 \times 10^{-15}\mbox{W}$$

...maybe its a nanobulb.

'''5. The photons in a beam of electromagnetic radiation carry 2.5μJ of energy each. How long should the phasors representing this radiation take to rotate?'''

First calculate the frequency of each photon:

$$f = \frac{E}{h} = \frac{2.5 \times 10^{-6}}{6.626 \times 10^{-34}} \approx 3.77 \times 10^{27}\mbox{Hz}$$ (That's one nasty gamma ray.)

Then calculate the time taken for one 'wavelength' to go by:

$$f = \frac{1}{t}$$

$$t = \frac{1}{f} = \frac{1}{3.77 \times 10^{27}} \approx 2.65 \times 10^{-28}\mbox{s}$$