Photons.

Step 1 of 7

A puzzle

In the light Unit we saw that light is an electromagnetic wave. In the wave picture, brighter light shakes the electrons in a metal harder, so light that is bright enough, of any colour, should knock electrons out of it.

The Scene shows a photocell: a sodium plate in a tube emptied of air, a ring that collects the electrons that leave it, and a current meter. The plate is lit with Red light, and Light power is in the middle. Predict what will happen when the light is turned up to 100 µW.

Step 2 of 7

Light arrives as photons

In 1905 Einstein proposed that light arrives in packets of energy, photons. A photon’s energy depends only on the frequency:

$E = hf$⁠

Here $h$⁠ is Planck’s constant, $h \approx 6.63 \times 10^{-34}\ \mathrm{J \cdot s}$$h \approx$$6.63 \times 10^{-34}\ \mathrm{J \cdot s}$⁠. These energies are very small, so they are measured in electronvolts: the energy an electron gains crossing a potential difference of one volt, 1 eV = 1.6 × 10⁻¹⁹ J.

Show the photons has shown the beam as photons. The light is now Blue, at 650 THz, that is $6.5 \times 10^{14}$⁠ hertz. Work out the energy of one photon and compare it with the readings beside the Scene: 2.69 eV.

Step 3 of 7

The threshold frequency

An electron in a metal is bound to it. Getting it out takes a minimum energy, the work function $W$⁠. A photon gives all its energy to one electron, all at once, and an electron cannot collect energy from several photons. So electrons leave only if $hf > W$⁠. The threshold frequency is the frequency at which one photon pays exactly the work function:

$f_0 = \frac{W}{h}$⁠

Show the photons now shows one photon’s energy budget. Metal plate is locked on Calcium, whose work function is 2.9 eV.

Step 4 of 7

The stopping voltage

Above the threshold, the electron uses $W$⁠ to get out and keeps the rest as kinetic energy. The fastest electrons are left with

$K_{\max} = hf - W$⁠

To measure $K_{\max}$⁠, a reverse voltage is connected that pushes the electrons back. The stopping voltage $V_s$⁠ is the voltage that stops even the fastest, and the current falls to zero. A voltage of one volt takes one electronvolt from an electron, so $eV_s = K_{\max}$⁠: the stopping voltage in volts equals $K_{\max}$⁠ in electronvolts. Here it is Blue light on Sodium, W = 2.3 eV.

Step 5 of 7

Brightness

If light is photons, what does brighter light change? The light power $P$⁠ is the energy arriving each second, so the number of photons each second is $\dfrac{P}{hf}$⁠. Light frequency is locked. Predict what will happen when Light power is doubled.

Step 6 of 7

The electron’s speed

An electron’s kinetic energy is $K = \tfrac12 m v^2$⁠, with the electron’s mass, $m = 9.11 \times 10^{-31}\ \mathrm{kg}$$m =$$9.11 \times 10^{-31}\ \mathrm{kg}$⁠. To substitute, first convert $K_{\max}$⁠ from electronvolts to joules. The light is now ultraviolet, 1000 THz, on sodium.

Step 7 of 7

Measuring Planck’s constant

The graph of $K_{\max}$⁠ against $f$⁠ is a straight line: $K_{\max} = hf - W$⁠. Its slope is $h$⁠, and it crosses the frequency axis at the threshold frequency. Change Metal plate: each metal moves the line along the axis, but the slope is the same for all of them. This is how Robert Millikan measured $h$⁠ in 1916, and it was the sign that photons are real.