A teacher charges a clean zinc plate with extra electrons and mounts it on an electroscope, so the two gold leaves stand apart. She carries it into a roadside welding shop, a metre from the blue-white arc. Within a second or two the leaves fall together. Back in the school hall, she holds the same charged plate under a bright red stage lamp for an hour. The leaves never move.
Electrons that belong to the metal
A copper wire carries a current because some of its electrons are free. Each copper atom gives up an outer electron, and these electrons wander through the whole piece of metal. They are called free electrons. Yet a copper wire lying on a table does not leak electrons into the room. Something holds them in.
The hold comes from the positive ions of the metal. Inside the metal, an electron has ions on every side, and their pulls balance. At the surface, an electron that starts to leave has ions only behind it. Their pull is no longer balanced, and it drags the electron back. So an electron is free inside the metal but meets a wall at its surface.
The height of the wall
Think of it in terms of energy, as in the chapter on Potential Energy. Take an electron at rest far outside the metal, and call its energy zero. The most loosely held electrons inside the metal have less energy than that. They sit at \(-\phi\), a depth \(\phi\) below the zero. To escape, such an electron must be given at least \(\phi\).
The work function \(\phi\) of a metal is the least energy that frees an electron from the surface of the metal.
The word "least" matters. Most electrons in the metal are held more deeply, and they need more than \(\phi\). The work function belongs to the most loosely held ones. An electron that is given more than it needs leaves with the extra energy as kinetic energy. So an electron given energy \(E\) leaves with at most \(E - \phi\).
Play Fig 1.1 for each metal. The electron always rises by the same \(3.00\,\text{eV}\). What changes is the depth it starts from. From caesium it clears the rim and leaves. From platinum it rises only part of the way, and the energy it took ends up warming the metal.
The work function of copper is \(4.65\,\text{eV}\). What does this number mean?
In joules these energies are awkward. The work function of caesium is \(3.42 \times 10^{-19}\,\text{J}\), and that of platinum is \(9.04 \times 10^{-19}\,\text{J}\). Every energy in this chapter is of this tiny size. A unit of the right size makes the arithmetic far easier.
The electron volt
The unit comes from the volt, so first a word about it. Between two points with a potential difference \(V\), a charge \(q\) that moves from one point to the other gains or loses kinetic energy \(qV\). The chapter on electromagnetism builds this idea properly. Here we need only that one fact.
The electron has the smallest charge found free in nature, \(e = 1.6 \times 10^{-19}\,\text{C}\). An electron that crosses a potential difference of \(1\,\text{V}\) gains \(1.6 \times 10^{-19}\,\text{J}\). This amount of energy is called one electron volt, written eV. An electron that crosses \(V\) volts gains \(V\) electron volts.
Now the work functions read as small, friendly numbers. The table below gives five metals; it will be used through the chapter.
- caesium, \(2.14\,\text{eV}\)
- potassium, \(2.30\,\text{eV}\)
- sodium, \(2.75\,\text{eV}\)
- copper, \(4.65\,\text{eV}\)
- platinum, \(5.65\,\text{eV}\)
The alkali metals, caesium, potassium and sodium, hold their outer electron loosely. Copper and platinum hold theirs about twice as tightly.
An electron starts from rest and crosses a potential difference of \(100\,\text{V}\). Find its kinetic energy in eV and in joules, and its speed.
(a) It crosses \(100\,\text{V}\), so it gains \(K = 100\,\text{eV}\).
(b) In joules, \(K = 100 \times 1.6 \times 10^{-19} = 1.6 \times 10^{-17}\,\text{J}\).
(c) From \(K = \tfrac12 m_e v^2\), \(v = \sqrt{2K/m_e} = \sqrt{2 \times 1.6 \times 10^{-17} / 9.1 \times 10^{-31}} = 5.9 \times 10^{6}\,\text{m/s}\), about 2 % of the speed of light.
Values used: \(e = 1.6 \times 10^{-19}\,\text{C}\), \(m_e = 9.1 \times 10^{-31}\,\text{kg}\).
The electron volt is a unit of energy, not of potential. An electron with \(3\,\text{eV}\) has an energy. Plates at \(3\,\text{V}\) have a potential difference. The two meet only in \(K = eV\).
An electron inside a metal absorbs \(4.8 \times 10^{-19}\,\text{J}\). How much energy is this in electron volts?
An electron and a proton each start from rest and cross the same \(12\,\text{V}\) battery, each in the direction that speeds it up. Which statement is correct?
Three ways to pay the cost
An electron leaves a metal whenever it gains at least the work function. The energy can come from three sources. Each gives its own kind of electron emission.
In thermionic emission the metal is heated. The free electrons share the random energy of heat, and at a high temperature a few of them have more than \(\phi\). These few escape. The glowing filament in the electron gun of an old television worked this way, and so does the filament of an X-ray tube.
In field emission a very strong electric field is applied at the surface, usually at a sharp metal tip. The field pulls electrons out of the metal even when it is cold. Some electron microscopes use such a tip to make a fine, bright beam.
In photoelectric emission, light falls on the metal and supplies the energy. The electrons it frees are called photoelectrons. This effect, the photoelectric effect, is the subject of this chapter.
The photoelectric effect is the emission of electrons from the surface of a metal when light of a suitable frequency falls on it.
The filament of an X-ray tube glows white-hot and releases electrons in complete darkness. Which kind of emission is this?
Back to the zinc plate
Now the first half of the puzzle has an answer. The welder's arc gives out ultraviolet light as well as visible light. That light frees electrons from the zinc plate. The plate is negative, so it pushes each freed electron away. The extra electrons leave one by one, and the leaves fall together.
A plate charged the other way behaves differently. If the zinc plate is positive, the light still frees electrons, but the positive plate pulls each of them straight back. So a positive plate keeps its charge under the arc. The electroscope tells us which charge light can remove: negative charge, carried by electrons.
A zinc plate on an electroscope is given a positive charge and held near a welding arc. What happens to the leaves?
The second half of the puzzle is harder. The red stage lamp is far brighter to the eye than the arc. It pours more energy onto the plate every second. Yet it frees no electrons at all, in a second or in an hour. Energy alone is clearly not the whole story. Lesson 3 finds the missing piece.
In 1888 the German physicist Wilhelm Hallwachs repeated a strange observation of Heinrich Hertz with a simpler instrument. He fixed a clean zinc plate to an electroscope and shone ultraviolet light from an arc lamp on it. A negatively charged plate lost its charge quickly. A positively charged plate did not, and a neutral plate became positively charged. The electron was not yet known, so Hallwachs could only say that light removes negative charge. Nine years later, J. J. Thomson identified the carrier of that charge as the electron.
- Taking the work function as the energy every electron needs.It is the least energy, for the most loosely held electron. Deeper electrons need more.
- Treating the electron volt as a unit of potential.eV is energy, \(1.6 \times 10^{-19}\,\text{J}\). V is potential difference. They meet in \(K = eV\).
- Thinking that an electron gets the work function as kinetic energy.The work function is paid on the way out. Only the energy beyond \(\phi\) remains as kinetic energy.
- Calling the glow of a hot filament the cause of its emission.Heat frees the electrons; the glow is a side effect. That is thermionic, not photoelectric, emission.