Dual Nature of Radiation and Matter — Part 2: Planck’s constant, photon momentum and matter waves: visual lesson, ISC Class 12 Physics

Lesson

Planck’s Constant and Matter Waves

Part 2 — measuring h from a straight-line graph, photons with momentum, and electrons that behave as waves.

In this lesson you will

Find Planck’s constant and the work function from a graph of stopping potential against frequency
Calculate the momentum of a photon
Calculate the de Broglie wavelength of an electron accelerated through a potential difference, and describe the Davisson–Germer experiment

barium, W₀ = 2.5 eV (f₀ = 6.06 × 10¹⁴ Hz)
magnesium, W₀ = 3.7 eV (f₀ = 8.97 × 10¹⁴ Hz)

Same slope h/e for every metal; the line starts at the threshold frequency f₀ = W₀/h.

Diagram

Stopping potential against frequency

barium, W₀ = 2.5 eV (f₀ = 6.06 × 10¹⁴ Hz)
magnesium, W₀ = 3.7 eV (f₀ = 8.97 × 10¹⁴ Hz)

Same slope h/e for every metal; the line starts at the threshold frequency f₀ = W₀/h.

Divide Einstein’s equation by e: $V_0 = \dfrac{h}{e}\,f - \dfrac{W_0}{e}$ A graph of $V_0$ against $f$ is a straight line with slope $\dfrac{h}{e}$ , the same for every metal, cutting the $f$ axis at $f_0$ .

With $h = 6.6 \times 10^{-34}$ J s the slope is $\dfrac{h}{e} = 4.125 \times 10^{-15}$ V s, that is 0.4125 V for every $10^{14}$ Hz.
Lines for different metals are parallel; a larger W₀ shifts the line to the right.

Concept

Reading h and W₀ from the graph

From a measured line:

Slope $= \dfrac{\Delta V_0}{\Delta f} = \dfrac{h}{e}$ , so $h = e \times \text{slope}$ .
Intercept on the f axis $= f_0$ , so $W_0 = hf_0$ .
Intercept on the $V_0$ axis (extended back to $f = 0$ ) $= -\dfrac{W_0}{e}$ .
Millikan did this carefully in 1916 and confirmed Einstein’s equation and the value of h.

barium, W₀ = 2.5 eV (f₀ = 6.06 × 10¹⁴ Hz)
magnesium, W₀ = 3.7 eV (f₀ = 8.97 × 10¹⁴ Hz)

Same slope h/e for every metal; the line starts at the threshold frequency f₀ = W₀/h.

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