Energy and Momentum of a Photon
Section: Quantum Physics | Syllabus: Cambridge AS Level Physics 9702
Particulate Nature of Electromagnetic Radiation Electromagnetic radiation consists of self-propagating oscillating electric and magnetic fields. While interference and diffraction provide evidence for a wave model, the photoelectric effect provides evidence for a particulate nature.
Photon A quantum of energy of electromagnetic radiation. Photons are massless but carry energy and momentum. Photon Energy The energy of a photon is directly proportional to the frequency of the radiation.
Photon Energy: E = hf where: E = photon energy (J) h = Planck constant (6.63 × 10^-34 J s) f = frequency (Hz) Substituting the wave equation c = fλ, we can write: E = hc/λ This shows that higher frequency (or shorter wavelength) radiation (like gamma rays) has higher energy photons than lower frequency radiation (like radio waves).
Ionising Radiation High-energy photons (UV, X-rays, Gamma) can remove electrons from atoms, creating ions. This is why they are dangerous to living tissue. Worked Examples Worked Example 1: Energy of Blue Light Question: Calculate the energy of a photon of blue light with a wavelength of 450 nm.
Solution 1. Convert wavelength to metres: λ = 450 × 10^-9 m 2. Use E = hc/λ: E = (6.63 × 10^-34)(3.00 × 10^8)450 × 10^-9 E = 4.42 × 10^-19 J Worked Example 2: Photons from a Laser Question: A laser pointer emits red light (λ = 650 nm) with a power of 3.0 mW.
Calculate the number of photons emitted per second. Solution 1. Energy of one photon: E = hc/λ = (6.63 × 10^-34)(3.00 × 10^8)650 × 10^-9 = 3.06 × 10^-19 J 2. Power P = Total Energy / time = N × E (where N is number of photons/sec).
3. N = P/E = 3.0 × 10^-33.06 × 10^-19 Answer: 9.8 × 10^15 s^-1 Worked Example: Conversions Question: (a) Convert 5.0 MeV to Joules. (b) Convert 8.0 × 10^-13 J to eV. Solution (a) 5.0 MeV = 5.0 × 10^6 eV E = (5.0 × 10^6) × (1.60 × 10^-19) = 8.0 × 10^-13 J (b) E = 8.0 × 10^-131.60 × 10^-19 = 5.0 × 10^6 eV = 5.0 MeV Worked Example: Momentum of a Gamma Ray Question: A gamma photon has energy 1.0 MeV.
Calculate its momentum. Solution 1. Convert E to Joules: 1.0 × 10^6 × 1.60 × 10^-19 = 1.60 × 10^-13 J 2. Use p = E/c: p = 1.60 × 10^-133.00 × 10^8 = 5.3 × 10^-22 kg m s^-1 Worked Example: Photon Energy from Frequency Question: Calculate the energy of a photon of red light with frequency 4.30 × 10^14 Hz.
Solution Using the photon energy equation E = hf: E = (6.63 × 10^-34)(4.30 × 10^14) E = 2.85 × 10^-19 J Worked Example: Energy from Photon Momentum Question: Calculate the energy of a photon with momentum 8.65 × 10^-26 kg m s^-1.
Give your answer in electronvolts. Solution 1. Using E = pc: E = (8.65 × 10^-26)(3.00 × 10^8) = 2.60 × 10^-17 J 2. Convert to electronvolts: E = 2.60 × 10^-171.60 × 10^-19 = 163 eV Electronvolt (eV) The joule is a very large unit for atomic-scale energies.
A more convenient unit is the electronvolt. Electronvolt (eV) The energy transferred to or from an electron when it moves through a potential difference of 1 volt. 1 eV = 1.60 × 10^-19 J Common multiples: 1 keV = 10^3 eV 1 MeV = 10^6 eV 1 GeV = 10^9 eV Photon Momentum Although photons have no mass, they have momentum.
This was a surprising result of quantum theory. p = E/c Substituting E=hf: p = hf/c = h/λ Units of Momentum SI unit: kg m s^-1 or N s. Common Misconception It is often thought that because p=mv, a massless particle must have zero momentum.
This is only true in classical mechanics. In relativity/quantum physics, energy and momentum are linked even for massless particles.
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