SPH4U Photons and the Photoelectric Effect: Notes and Practice

Key things to remember

1Photon energy:

E = h f = \frac{h c}{λ}

, with

h = 6.63 \times 1 0^{- 34} J \cdot s

1 eV = 1.60 \times 1 0^{- 19} J

; divide joules by

1.60 \times 1 0^{- 19}

to get electron-volts.

3Photoelectric equation:

E_{k, ma x} = h f - W

, and the threshold frequency satisfies

W = h f_{0}

4Stopping potential:

E_{k, ma x} = e V_{s}

, so an energy of

x

eV is stopped by

x

volts.

5Intensity changes the number of electrons emitted; only frequency changes their maximum kinetic energy.

By 1900 the wave model of light looked unbeatable: it explained interference, diffraction, polarization and the double-slit experiment you studied last unit. Yet one simple experiment refused to fit. Shine light on a clean metal surface and electrons can fly off it — but a dim violet lamp ejects electrons while a blindingly bright red one ejects none at all. No wave explanation could account for that. The explanation that did work started the quantum revolution.

Energy comes in packets: the photon

In 1900 Max Planck found he could only match the measured colours of glowing hot objects by assuming energy is emitted in discrete amounts, or quanta, proportional to frequency. In 1905 Albert Einstein took the idea further: light itself travels as a stream of energy packets, now called photons. Each photon carries an energy set entirely by its frequency. A brighter beam of the same colour has more photons, not bigger photons.

E = h f = \frac{h c}{λ} h = 6.63 \times 1 0^{- 34} J \cdot s 1 eV = 1.60 \times 1 0^{- 19} J

Photon energy depends only on frequency or wavelength. The electron-volt is a convenient energy unit at atomic scale.

The photoelectric experiment

In a phototube, light falls on a metal plate (the emitter) inside an evacuated glass tube. Any electrons knocked out — called photoelectrons — cross to a collector and register as a current on an ammeter. To measure how much kinetic energy the fastest electrons have, the collector is made negative, so it repels them. The reverse voltage is raised until even the fastest electron is turned back and the current just drops to zero. That voltage is the stopping potential $V_{s}$ , and the work it does on the electron equals the electron's maximum kinetic energy: $E_{k, ma x} = e V_{s}$ .

Observation	What the wave model predicts	What actually happens
Light below a certain frequency	Any colour will eject electrons if it is bright enough	No electrons at all below the threshold frequency $f_{0}$ , however bright
Increasing the intensity	Electrons should leave with more energy	More electrons leave (larger current), but the maximum kinetic energy is unchanged
Increasing the frequency	Should make no particular difference	Maximum kinetic energy increases, in a straight line with frequency
Very dim light above threshold	A delay while electrons soak up enough energy	Electrons are emitted essentially instantly

Einstein's photoelectric equation

One photon gives all its energy to one electron. Part of that energy, the work function

W

, is used to free the electron from the metal; whatever is left becomes kinetic energy. Electrons deeper in the metal lose a little more on the way out, so the equation gives the maximum kinetic energy. If

h f

is less than

W

, no single photon can free an electron, and adding more such photons does not help.

E_{k, ma x} = h f - W W = h f_{0} E_{k, ma x} = e V_{s}

The threshold frequency f0 is the frequency whose photons have just enough energy to free an electron.

Fig 1Maximum kinetic energy against frequency for two metals. Both lines have slope h; each crosses the frequency axis at its own threshold frequency.

The graph of $E_{k, ma x}$ against $f$ is the fingerprint of the photon model. Every metal gives a straight line with the same slope, $h$ , because the slope is a property of light, not of the metal. Each line crosses the frequency axis at its threshold frequency $f_{0}$ , and extending it back to $f = 0$ gives a vertical intercept of $- W$ . Robert Millikan set out to test Einstein's equation with careful measurements of stopping potentials and found that it held, with a slope matching Planck's constant.

Brighter is not faster

Intensity controls how many photons arrive each second, so it controls how many electrons are ejected — the current. Frequency controls the energy of each photon, so it controls how fast the electrons leave. A student who says 'brighter light gives electrons more energy' is using the wave model this experiment disproved.

Loading interactive…

Choose a metal, then change the frequency and the intensity separately and watch which one changes the electron energy.

The photon model did not replace the wave model — interference is still real. Instead, light behaves as a wave when it travels and spreads out, and as a stream of particles when it is emitted or absorbed. Devices that count individual photons or turn light directly into an electric signal, such as photomultiplier tubes and the image sensors in digital cameras, all depend on light releasing electrons one photon at a time. Einstein's Nobel Prize in Physics was awarded for this explanation of the photoelectric effect.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

Find the energy of a photon of green light with wavelength

500 nm

, in joules and in electron-volts.

Example 2

Violet light of wavelength

400 nm

shines on a metal with a work function of

2.30 eV

. Find the maximum kinetic energy of the photoelectrons (in eV and J) and the stopping potential.

Example 3

A metal has a work function of

4.30 eV

. Find its threshold frequency and the longest wavelength of light that can eject electrons from it.

Example 4

Light of frequency

1.00 \times 1 0^{15} Hz

ejects electrons from a metal, and a stopping potential of

1.20 V

is needed to stop them. Find the work function of the metal in electron-volts.

Where marks get lost

Subtracting a work function in electron-volts from a photon energy in joules. Convert both to the same unit before using

E_{k, ma x} = h f - W

Saying that brighter light makes the electrons faster. Intensity changes the number of electrons; only frequency changes their maximum kinetic energy.

Getting a negative kinetic energy and reporting it. A negative result means the photon energy is below the work function, so no electrons are emitted at all.

Forgetting to convert nanometres to metres in

E = h c / λ

, which makes the energy

1 0^{9}

times too small.

Mixing up the threshold frequency and the threshold wavelength: light must be above

f_{0}

, which means below

λ_{0}

Key things to remember

Worked examples

Where marks get lost

Flashcards 8

Practice