Photon Energy Calculations

In the last lesson we introduced the photon model and the Planck–Einstein relation E = hf = hc/λ. You should now be comfortable with the idea that electromagnetic energy is quantised. This lesson is about doing quantum arithmetic: taking a wavelength or a frequency or an energy and computing the other two, converting between joules and electronvolts, and comparing photons across the electromagnetic spectrum. OCR exam questions rarely allow you to use the Planck relation in only one direction, and they frequently combine it with other formulae from the specification — conservation of energy, c = fλ, and the definition of the electronvolt.

The material in this lesson is methodological rather than conceptual. It gives you a toolkit of worked examples to draw on whenever the exam asks for a numerical answer.

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Recap: The Core Equations

Before we begin calculating, let us collect the equations in one place. You will need:

Quantity	Equation	Units
Photon energy (from frequency)	E = hf	J
Photon energy (from wavelength)	E = hc/λ	J
Wave equation	c = fλ	—
Energy conversion	1 eV = 1.60 × 10⁻¹⁹ J	—
Number of photons from total energy	N = E_total/E_photon	—

The constants, given in the OCR data sheet, are:

h = 6.63 × 10⁻³⁴ J s
c = 3.00 × 10⁸ m s⁻¹
e = 1.60 × 10⁻¹⁹ C

It is worth memorising the product hc = 1.99 × 10⁻²⁵ J m. This comes up so often in calculations that having it to hand saves time.

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Worked Example 1: Visible Light

A photon of blue light has wavelength 450 nm. Calculate its energy in joules and in electronvolts.

Solution.

Step 1. Use E = hc/λ, taking care with units.

E = hc/λ
  = (1.99 × 10⁻²⁵ J m)/(450 × 10⁻⁹ m)
  = 4.42 × 10⁻¹⁹ J

Step 2. Convert to eV by dividing by e:

E = (4.42 × 10⁻¹⁹)/(1.60 × 10⁻¹⁹) eV = 2.76 eV

Answer: E ≈ 4.4 × 10⁻¹⁹ J or 2.8 eV.

Exam Tip: Notice how quickly the calculation proceeds once you have memorised hc = 1.99 × 10⁻²⁵ J m. If you compute (6.63 × 10⁻³⁴)(3.00 × 10⁸) every time, you are doing unnecessary work and risking arithmetic mistakes.

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Worked Example 2: Infrared Radiation

An infrared photon has frequency 3.75 × 10¹³ Hz. What is its energy (a) in joules, (b) in electronvolts, and (c) what is its wavelength?

Solution.

(a) E = hf = (6.63 × 10⁻³⁴)(3.75 × 10¹³) = 2.49 × 10⁻²⁰ J.

(b) E = (2.49 × 10⁻²⁰)/(1.60 × 10⁻¹⁹) = 0.155 eV.

(c) λ = c/f = (3.00 × 10⁸)/(3.75 × 10¹³) = 8.00 × 10⁻⁶ m = 8.00 μm.

This is mid-infrared, the characteristic wavelength of thermal radiation from objects near room temperature.

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Worked Example 3: Comparing Red and Violet Light

Red light has wavelength 700 nm. Violet light has wavelength 400 nm. Calculate the ratio of the photon energies E_violet : E_red.

Solution. Because E = hc/λ, photon energy is inversely proportional to wavelength. Hence

E_violet/E_red = λ_red/λ_violet
               = 700/400
               = 1.75

A violet photon carries 1.75× the energy of a red photon. Numerically:

E_red = (1.99 × 10⁻²⁵)/(700 × 10⁻⁹) = 2.84 × 10⁻¹⁹ J = 1.78 eV
E_violet = (1.99 × 10⁻²⁵)/(400 × 10⁻⁹) = 4.98 × 10⁻¹⁹ J = 3.11 eV

and indeed 3.11/1.78 ≈ 1.75.

Exam Tip: OCR questions frequently ask for ratios of photon energies across different wavelengths. Notice how the ratio comes out directly from the inverse relationship, without needing to calculate either absolute value separately. This is often the quickest route through the calculation.

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Worked Example 4: Counting Photons in a Light Beam

A 5.0 mW laser produces green light of wavelength 532 nm. How many photons does the laser emit per second?

Solution.

Step 1. Calculate the energy of a single photon.

E_photon = hc/λ
         = (1.99 × 10⁻²⁵)/(532 × 10⁻⁹)
         = 3.74 × 10⁻¹⁹ J

Step 2. The power P = 5.0 mW means the laser delivers 5.0 × 10⁻³ J per second. The number of photons per second is

N/t = P/E_photon
    = (5.0 × 10⁻³)/(3.74 × 10⁻¹⁹)
    = 1.34 × 10¹⁶ photons/s

Answer: The laser emits about 1.3 × 10¹⁶ photons every second.

This is an enormous number, which is why the beam looks continuous. Our eyes can no more count individual photons than we can count the raindrops in a downpour.

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Worked Example 5: Photon Energy at an X-Ray Wavelength

A medical X-ray source produces photons of wavelength 5.0 × 10⁻¹¹ m (0.05 nm). Calculate the photon energy in keV.

Solution.

E = hc/λ
  = (1.99 × 10⁻²⁵)/(5.0 × 10⁻¹¹)
  = 3.98 × 10⁻¹⁵ J

In electronvolts:

E = (3.98 × 10⁻¹⁵)/(1.60 × 10⁻¹⁹) eV = 2.49 × 10⁴ eV = 24.9 keV

A 25 keV photon is characteristic of a diagnostic X-ray machine — enough energy to penetrate soft tissue, ionise atoms, and produce a radiographic image.

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Worked Example 6: Reverse Problem

A photon has energy 3.40 eV. What is its wavelength?

Solution. First convert the energy to joules: