Numford
Photoelectric

Einstein's photoelectric equation

Einstein's photoelectric equation is K max = hν - φ. For 400 nm light on potassium (φ = 2.3 eV), hν = 3.11 eV, so K max = 0.81 eV and V₀ = 0.81 V.

Photoelectric effect

Pick a metal and a colour of light. Each photon carries hν; it frees an electron only if hν is more than the work function φ. Brighter light means more photons, not bigger ones. Values use h = 6.63 × 10⁻³⁴ J s and e = 1.6 × 10⁻¹⁹ C.

Metal
Light
Photon energy3.11 eV4.97 × 10⁻¹⁹ J
Threshold5.55 × 10¹⁴ Hzλ₀ = 540 nm
K max0.81 eVfastest at 5.33 × 10⁵ m/s
Stopping potential0.81 Vstops even the fastest
Photons per second4.0 × 10¹⁵at 2 mW

K plate in light of 400 nm: electrons leave with up to 0.81 eV.

5-22ν (10¹⁴ Hz)K max (eV)ν₀ = 5.55-φ = -2.3 eVslope h
K max against ν
-10123V (volt)Current2 mW1 mW-V₀
Current against voltage
Show the working
  1. ν = c / λ = 3 × 10⁸ / (400 × 10⁻⁹) = 7.5 × 10¹⁴ Hz
  2. E = hν = 6.63 × 10⁻³⁴ × 7.5 × 10¹⁴ = 4.97 × 10⁻¹⁹ J = 4.97 × 10⁻¹⁹ / 1.6 × 10⁻¹⁹ = 3.11 eV
  3. ν₀ = φ / h = 2.3 × 1.6 × 10⁻¹⁹ / 6.63 × 10⁻³⁴ = 5.55 × 10¹⁴ Hz, so λ₀ = c / ν₀ = 540 nm
  4. K max = hν - φ = 3.108 - 2.3 = 0.81 eV
  5. eV₀ = K max, so V₀ = 0.81 V
  6. v max = √(2K / m) = √(2 × 1.29 × 10⁻¹⁹ / 9.11 × 10⁻³¹) = 5.33 × 10⁵ m/s
  7. Photons per second = P / E = 2.0 × 10⁻³ W / 4.97 × 10⁻¹⁹ J = 4.0 × 10¹⁵
Working

Step by step

  1. ν = c / λ = 3 × 10⁸ / (400 × 10⁻⁹) = 7.5 × 10¹⁴ Hz
  2. E = hν = 6.63 × 10⁻³⁴ × 7.5 × 10¹⁴ = 4.97 × 10⁻¹⁹ J = 4.97 × 10⁻¹⁹ / 1.6 × 10⁻¹⁹ = 3.11 eV
  3. ν₀ = φ / h = 2.3 × 1.6 × 10⁻¹⁹ / 6.63 × 10⁻³⁴ = 5.55 × 10¹⁴ Hz, so λ₀ = c / ν₀ = 540 nm
  4. K max = hν - φ = 3.108 - 2.3 = 0.81 eV
  5. eV₀ = K max, so V₀ = 0.81 V
  6. v max = √(2K / m) = √(2 × 1.29 × 10⁻¹⁹ / 9.11 × 10⁻³¹) = 5.33 × 10⁵ m/s
  7. Photons per second = P / E = 2.0 × 10⁻³ W / 4.97 × 10⁻¹⁹ J = 4.0 × 10¹⁵
Derivation

Why it works

  1. Light arrives in packets, photons, each of energy E = hν.
  2. One photon is absorbed by one electron, all at once, so there is no delay however dim the light.
  3. The electron must spend at least the work function φ to leave the metal.
  4. The fastest electrons keep the rest: K max = hν - φ. This is Einstein's photoelectric equation.
  5. A retarding voltage V₀ stops even these, so eV₀ = K max. When hν = φ, K max = 0: that frequency is the threshold ν₀ = φ / h.
Questions

Einstein's photoelectric equation: quick answers

What does each term in K max = hν - φ mean?

hν is the energy of one photon, φ is the work function (the least energy needed to pull an electron out of the metal), and K max is the kinetic energy of the fastest electrons that leave.

Why is it K max and not just K?

Electrons deeper in the metal lose some energy on the way out, so they leave slower. Only electrons from the surface keep all of hν - φ.

What is K max for 400 nm light on potassium?

E = hc / λ = 3.11 eV, and φ = 2.3 eV, so K max = 0.81 eV. The fastest electrons move at 5.33 × 10⁵ m/s.

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