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Photoelectric

Photoelectric effect graphs

K max against frequency is a straight line of slope h = 6.63 × 10⁻³⁴ J s that cuts the axis at ν₀ (6.64 × 10¹⁴ Hz for sodium). Current against voltage saturates, and falls to zero at -V₀ = -0.56 V for light of 8.0 × 10¹⁴ Hz, whatever the intensity.

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 energy3.31 eV5.3 × 10⁻¹⁹ J
Threshold6.64 × 10¹⁴ Hzλ₀ = 452 nm
K max0.56 eVfastest at 4.45 × 10⁵ m/s
Stopping potential0.56 Vstops even the fastest
Photons per second3.8 × 10¹⁵at 2 mW

Na plate in light of 375 nm: electrons leave with up to 0.56 eV.

510-22ν (10¹⁴ Hz)K max (eV)ν₀ = 6.64-φ = -2.75 eVslope h
K max against ν
-10123V (volt)Current2 mW1 mW-V₀
Current against voltage
Show the working
  1. E = hν = 6.63 × 10⁻³⁴ × 8.0 × 10¹⁴ = 5.3 × 10⁻¹⁹ J = 5.3 × 10⁻¹⁹ / 1.6 × 10⁻¹⁹ = 3.31 eV
  2. ν₀ = φ / h = 2.75 × 1.6 × 10⁻¹⁹ / 6.63 × 10⁻³⁴ = 6.64 × 10¹⁴ Hz, so λ₀ = c / ν₀ = 452 nm
  3. K max = hν - φ = 3.315 - 2.75 = 0.56 eV
  4. eV₀ = K max, so V₀ = 0.56 V
  5. v max = √(2K / m) = √(2 × 9.04 × 10⁻²⁰ / 9.11 × 10⁻³¹) = 4.45 × 10⁵ m/s
  6. Photons per second = P / E = 2.0 × 10⁻³ W / 5.3 × 10⁻¹⁹ J = 3.8 × 10¹⁵
Working

Step by step

  1. Graph 1: K max = hν - φ is a straight line, y = mx + c.
  2. Slope = h = 6.63 × 10⁻³⁴ J s, the same for every metal.
  3. It cuts the ν axis at ν₀ = φ / h = 6.64 × 10¹⁴ Hz for sodium, and extended back it cuts the K axis at -φ = -2.75 eV.
  4. For light of 8.0 × 10¹⁴ Hz: K max = 3.31 eV - 2.75 eV = 0.56 eV, the glowing point.
  5. Graph 2: the current saturates when every electron reaches the collector. Half the light gives half the saturation current.
  6. Both curves reach zero at the same -V₀ = -0.56 V: V₀ depends on the frequency, not on the intensity.
Derivation

Why it works

  1. Write Einstein's equation as K max = hν - φ, or eV₀ = hν - φ.
  2. This is y = mx + c with y = K max and x = ν, so the graph is a straight line.
  3. Its slope is h, the same for every metal, so lines for different metals are parallel.
  4. It cuts the frequency axis at ν₀ = φ / h, and extended back it cuts the energy axis at -φ.
  5. The current against voltage graph saturates when every emitted electron is collected, and falls to zero at -V₀, which depends on ν but not on intensity.
Questions

Photoelectric effect graphs: quick answers

What is the slope of the K max against frequency graph?

Planck's constant h. If the graph is of stopping potential V₀ against frequency instead, the slope is h / e.

Why are the lines for different metals parallel?

Every metal has the same slope h. Only the intercept changes: each line cuts the frequency axis at that metal's own threshold frequency.

Why does the current not stop at zero volts?

Electrons leave with kinetic energy, so many reach the collector even with no voltage. A reverse voltage is needed to stop them, and at -V₀ even the fastest are turned back.

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