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 = h\nu = \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

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 = h\nu \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

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 \(E = h\nu = hc/\lambda\) 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.