understand that a photon has momentum and that the momentum is given by p = E / c

Energy and Momentum of a Photon – Cambridge IGCSE/A‑Level (9702) Syllabus

Learning Objective

Students will be able to:

• Define a photon and state that it is mass‑less but carries energy and momentum.
• Write the two equivalent forms of the photon‑energy relation \[ E = hu = \frac{hc}{\lambda} \] and use the electron‑volt (eV) as the preferred unit for photon energies.
• Derive and apply the photon‑momentum relation \[ p = \frac{E}{c} = \frac{h}{\lambda} \] in radiation‑pressure and solar‑sail problems.
• Use the photo‑electric equation \[ hf = \Phi + \frac12 mv_{\max}^{2} \] to calculate threshold frequency, work‑function and maximum kinetic energy of emitted electrons.

Mapping to Syllabus

Syllabus Item	Coverage in these notes
22.1 – Photon definition & energy relation \(E=hf=hc/λ\)	Explicit statement of both forms; energy in joules and eV.
22.1 – Momentum relation \(p=E/c = h/λ\)	Derivation from relativistic relation; worked example with unit conversion to N·s and macroscopic comparison.
22.1 – Use of \(p=h/λ\) in radiation‑pressure problems	Step‑by‑step calculation for an absorbing and a reflecting surface (solar‑sail example).
22.2 – Photo‑electric effect equation \(hf = Φ + \frac12 mv_{\max}^2\)	Full equation, threshold frequency, and a template for solving typical exam questions.

Key Concepts

• Photon: quantum (particle) of electromagnetic radiation; rest mass \(m_0 = 0\).
• Planck’s constant: \(h = 6.626\times10^{-34}\,\text{J·s}\) (or \(4.136\times10^{-15}\,\text{eV·s}\)).
• Speed of light: \(c = 3.00\times10^{8}\,\text{m·s}^{-1}\).
• Energy unit: 1 eV = \(1.602\times10^{-19}\) J (convenient for photon energies).
• Radiation pressure: force per unit area exerted when photons are absorbed or reflected.

Photon Energy

Two interchangeable forms:

\[ E = hu \qquad\text{and}\qquad E = \frac{hc}{\lambda} \]

When the wavelength is given, the second form is usually most convenient. Converting to electron‑volts:

\[ E(\text{eV}) = \frac{1240}{\lambda(\text{nm})} \]

Example: \(\lambda = 500\ \text{nm}\) (green light)

\[ E = \frac{1240}{500} = 2.48\ \text{eV} \]

Photon Momentum

From the relativistic energy–momentum relation \(E^{2} = (pc)^{2} + (m_{0}c^{2})^{2}\) with \(m_{0}=0\):

\[ E = pc \;\Longrightarrow\; p = \frac{E}{c} = \frac{h}{\lambda} \]

Worked Example – Momentum of a 500 nm Photon

1. Calculate \(p\) using \(p = h/\lambda\):
\[ p = \frac{6.626\times10^{-34}\ \text{J·s}}{5.00\times10^{-7}\ \text{m}} = 1.33\times10^{-27}\ \text{kg·m·s}^{-1} \]
2. Convert to newton‑seconds (1 kg·m·s⁻¹ = 1 N·s):
\[ p = 1.33\times10^{-27}\ \text{N·s} \]
3. Compare with a macroscopic object (baseball, \(m=0.145\ \text{kg}\), speed \(v=30\ \text{m·s}^{-1}\)):
\[ p_{\text{baseball}} = mv = 0.145 \times 30 = 4.35\ \text{N·s} \] \[ \frac{p_{\text{baseball}}}{p_{\text{photon}}} \approx 3.3\times10^{27} \] The photon’s momentum is astronomically smaller, explaining why a single photon’s effect is imperceptible.

Radiation‑Pressure Applications (Use of \(p = h/λ\))

General Formulae

• Perfect absorber: \(P = \dfrac{I}{c}\)
• Perfect reflector: \(P = \dfrac{2I}{c}\)

where \(I\) is the light intensity (W m⁻²) and \(c\) the speed of light.

Sample Calculation – Solar Sail

Given:

• Solar constant at 1 AU: \(I = 1360\ \text{W·m}^{-2}\)
• Sail area: \(A = 100\ \text{m}^{2}\)
• Reflecting sail (ideal mirror).

Step‑by‑step:

1. Radiation pressure on a perfect reflector:
   \(P = \dfrac{2I}{c} = \dfrac{2 \times 1360}{3.00\times10^{8}} = 9.07\times10^{-6}\ \text{N·m}^{-2}\).
2. Force on the sail:
   \(F = PA = 9.07\times10^{-6} \times 100 = 9.07\times10^{-4}\ \text{N}\).
3. Resulting acceleration for a spacecraft of mass \(m = 500\ \text{kg}\):
   \(a = F/m = 1.81\times10^{-6}\ \text{m·s}^{-2}\).

This tiny acceleration builds up over months to give a measurable change in velocity, illustrating why photon pressure is significant for space propulsion.

Photon Momentum for Different Radiation Types

Radiation	Wavelength \(\lambda\) (m)	Energy \(E\) (J)	Momentum \(p\) (N·s)
Visible (green, 500 nm)	5.0 × 10⁻⁷	3.98 × 10⁻¹⁹	1.33 × 10⁻²⁷
Ultraviolet (20 nm)	2.0 × 10⁻⁸	9.94 × 10⁻¹⁸	3.31 × 10⁻²⁶
X‑ray (0.1 nm)	1.0 × 10⁻¹⁰	1.99 × 10⁻¹⁵	6.63 × 10⁻²⁴
Gamma ray (0.01 nm)	1.0 × 10⁻¹²	1.99 × 10⁻¹³	6.63 × 10⁻²²

Photo‑electric Effect (Syllabus 22.2)

Fundamental Equation

\[ hf = \Phi + \frac12 mv_{\max}^{2} \]

• \(h\) – Planck’s constant.
• \(f\) – frequency of the incident photon.
• \(\Phi\) – work‑function (minimum energy to free an electron, usually given in eV).
• \(\frac12 mv_{\max}^{2}\) – maximum kinetic energy of the emitted electrons.

Threshold Frequency

\[ hf_{\text{th}} = \Phi \qquad\Longrightarrow\qquad f_{\text{th}} = \frac{\Phi}{h} \]

Step‑by‑Step Template for Exam Questions

1. Convert wavelength to frequency (or vice‑versa) using \(c = \lambda f\).
2. Calculate photon energy \(E = hf\) (or \(E = hc/\lambda\)). Convert to eV if \(\Phi\) is given in eV.
3. Compare with the work‑function \(\Phi\)**:
   • If \(E < \Phi\) → no electrons emitted.
   • If \(E \ge \Phi\) → proceed to next step.
4. Find maximum kinetic energy \(K_{\max}=E-\Phi\) (in joules or eV).
5. Obtain the maximum speed using \(K_{\max}= \tfrac12 mv_{\max}^{2}\) (optional).
6. State the **threshold frequency** or **threshold wavelength** if required.

Sample Problem & Solution

Problem: A metal has a work‑function \(\Phi = 2.1\ \text{eV}\). Light of wavelength \(\lambda = 400\ \text{nm}\) shines on the surface. Find the maximum kinetic energy of the emitted electrons (in eV) and the threshold wavelength for this metal.

Solution:

1. Convert wavelength to frequency: \(f = c/\lambda = 3.00\times10^{8}\ /\ 4.00\times10^{-7} = 7.50\times10^{14}\ \text{Hz}\).
2. Photon energy (use \(h = 4.136\times10^{-15}\ \text{eV·s}\)): \(E = hf = 4.136\times10^{-15}\times7.50\times10^{14}=3.10\ \text{eV}\).
3. Maximum kinetic energy: \(K_{\max}=E-\Phi = 3.10 - 2.1 = 1.0\ \text{eV}\).
4. Threshold wavelength: \(\lambda_{\text{th}} = hc/\Phi\). Using \(hc = 1240\ \text{eV·nm}\): \(\lambda_{\text{th}} = 1240 / 2.1 = 590\ \text{nm}\).

Result: \(K_{\max}=1.0\ \text{eV}\); \(\lambda_{\text{th}}=590\ \text{nm}\).

Other A‑Level Applications

• Radiation pressure on an absorbing surface: \(P = I/c\).
• Radiation pressure on a reflecting surface: \(P = 2I/c\).
• Solar‑sail thrust calculations (see example above).
• Photo‑electric calculations in kinetic‑energy diagrams and stopping‑potential problems.

Common Misconceptions

• “Photons have no momentum because they have no mass.” – Momentum for photons derives from their energy: \(p = E/c\). Rest mass is not required.
• Confusing \(p = E/c\) with \(p = mv\). – The latter applies only to particles with non‑zero rest mass.
• Radiation pressure is always negligible. – In vacuum (e.g., space) the continuous pressure from sunlight can produce measurable thrust over long periods.

Suggested Classroom Activities

1. Laser‑mirror recoil – Suspend a lightweight mirror by a fine thread; shine a continuous‑wave laser on it and measure the tiny displacement to demonstrate photon momentum transfer.
2. Solar‑sail design task – Provide the solar constant and ask students to compute thrust for different sail areas and for absorbing vs. reflecting materials.
3. Photo‑electric workshop – Use a set of photons (different wavelengths) and a metal with known \(\Phi\); students calculate \(K_{\max}\) and predict whether electrons are emitted.
4. Scale‑comparison discussion – Compare the momentum of a single photon with that of everyday objects (e.g., a baseball) to highlight the enormous magnitude gap.

Summary

Even though photons have zero rest mass, they carry energy \(E = hu = hc/\lambda\) and momentum \(p = E/c = h/\lambda\). These relations are essential for:

• Quantifying radiation pressure and designing solar‑sail propulsion.
• Explaining the photo‑electric effect and solving related exam problems.
• Understanding why photon momentum is tiny compared with macroscopic objects, yet becomes significant in space‑based applications.

Mastery of these concepts fulfills the Cambridge International AS & A Level Physics (9702) requirements for syllabus points 22.1 and 22.2.