A photon’s energy depends on its frequency: E = hf. Since c = fλ, the same energy is E = hc ÷ λ, so a shorter wavelength gives more energy.
This lesson follows explaining the photoelectric effect. The wave speed idea is the same as in relating speed, frequency and wavelength.
Which formula do I start with?
Look at what the question gives. If it gives frequency, use E = hf directly. If it gives wavelength, either find f = c ÷ λ first, or use E = hc ÷ λ in one step.
Worked example: green light, two routes
An original green light has wavelength 5.5 × 10⁻⁷ m.
Route 1. Frequency f = c ÷ λ = 3.0 × 10⁸ ÷ 5.5 × 10⁻⁷ = 5.5 × 10¹⁴ Hz. Then E = hf = 6.63 × 10⁻³⁴ × 5.5 × 10¹⁴ = 3.6 × 10⁻¹⁹ J.
Route 2. E = hc ÷ λ = (6.63 × 10⁻³⁴ × 3.0 × 10⁸) ÷ (5.5 × 10⁻⁷) = 1.989 × 10⁻²⁵ ÷ (5.5 × 10⁻⁷) = 3.6 × 10⁻¹⁹ J.
Both routes agree, which confirms the answer.
How do the colours compare?
Using hc = 1.989 × 10⁻²⁵ J m, compare three original wavelengths.
| Light | Wavelength (m) | Photon energy (J) |
|---|---|---|
| Ultraviolet | 3.0 × 10⁻⁷ | 6.6 × 10⁻¹⁹ |
| Green | 5.5 × 10⁻⁷ | 3.6 × 10⁻¹⁹ |
| Red | 7.0 × 10⁻⁷ | 2.8 × 10⁻¹⁹ |
As wavelength increases down the table, photon energy decreases.
The mistake that costs marks
The slip is to put the wavelength into E = hf as if it were a frequency.
| Step | Wrong | Right |
|---|---|---|
| Substitute for green light | 6.63 × 10⁻³⁴ × 5.5 × 10⁻⁷ | Find f first, or use hc ÷ λ |
| Result | 3.6 × 10⁻⁴⁰ J | 3.6 × 10⁻¹⁹ J |
A quick check: visible-light photons have energies around 10⁻¹⁹ J. An answer of 10⁻⁴⁰ J signals that wavelength went into the wrong slot.
How many photons does a light source emit?
Divide the power by the energy of one photon. An original lamp gives out 1.0 W of green light, where each photon has 3.6 × 10⁻¹⁹ J.
Photons per second = 1.0 ÷ (3.6 × 10⁻¹⁹) = 2.8 × 10¹⁸. The number is enormous because each photon carries very little energy.
Check yourself
A photon has energy 4.0 × 10⁻¹⁹ J. Find its frequency and its wavelength.
Answer
f = E ÷ h = 4.0 × 10⁻¹⁹ ÷ 6.63 × 10⁻³⁴ = 6.0 × 10¹⁴ Hz.
λ = c ÷ f = 3.0 × 10⁸ ÷ 6.0 × 10¹⁴ = 5.0 × 10⁻⁷ m.
Check with hc ÷ E = 1.989 × 10⁻²⁵ ÷ 4.0 × 10⁻¹⁹ = 5.0 × 10⁻⁷ m.
What to study next
Photon energy is only part of the picture. A graph of electron energy against frequency shows both h and the work function. Continue with reading threshold-frequency graphs.
For help with the order of steps, see online one-to-one Physics tuition.