Electric current in various media

An electric current is an ordered movement of charged particles. But these particles differ from one substance to another: in metals they are free electrons, in electrolytes ions, in gases ions and electrons, in a vacuum electrons released from an electrode, and in semiconductors electrons and holes. Knowing the type of charge carrier explains how resistance depends on temperature and how each medium is used in technology. For every medium we answer three questions: where the carriers come from, how they move, and how the resistance changes on heating.

§1. Current in metals

In a metal’s crystal lattice the outer (valence) electrons break away from their atoms and form a free electron gas that moves throughout the sample; positive ions remain at the lattice sites. In Riecke’s experiment a current was passed for a year through copper and aluminium cylinders, yet their mass and composition did not change — so in a metal the current is carried by electrons, not ions.

Without a field the electrons move randomly. When a field is applied, a small ordered shift — drift — is added, and this is the current:

where n is the concentration of free electrons, v the drift velocity and S the cross-sectional area. The drift velocity is tiny (fractions of a millimetre per second), yet a lamp lights at once, because the field is set up along the whole wire almost at the speed of light and all the electrons start moving together.

Electrons collide with the lattice ions; when the metal is heated, the ions vibrate more strongly and collisions become more frequent, so a metal’s resistance increases:

• R₀ is the resistance at 0 °C, α the temperature coefficient of resistance (K⁻¹), about 4·10⁻³ K⁻¹ for pure metals.
• Resistance thermometers are based on this relationship.
• Some metals lose all resistance at very low temperatures — superconductivity (at 4.2 K for mercury). Superconducting coils produce the strong magnetic field in MRI scanners.

Worked example

A copper wire has a resistance of 10 Ω at 0 °C, α = 4.3·10⁻³ K⁻¹. What is its resistance at 100 °C?

1. R = R₀(1 + αt) = 10·(1 + 4.3·10⁻³·100) = 10·1.43.
2. R = 14.3 Ω.

Worked example

A current of 1 A flows in a copper wire of cross-section 1 mm², n = 8.5·10²⁸ m⁻³. What is the drift velocity of the electrons?

1. v = I/(neS) = 1/(8.5·10²⁸·1.6·10⁻¹⁹·10⁻⁶).
2. v ≈ 7.4·10⁻⁵ m/s — only 0.07 mm per second.

§2. Current in electrolytes and electrolysis

In aqueous solutions (and melts) of salts, acids and alkalis, molecules split into positive and negative ions — electrolytic dissociation. In a field, cations (positive ions) move to the cathode (−) and anions (negative ions) to the anode (+), and substances are released at the electrodes — electrolysis. Here matter is transported along with the current. The resistance of an electrolyte decreases on heating: dissociation increases, the solution becomes less viscous and the ions move faster.

• k is the electrochemical equivalent (kg/C), M the molar mass (kg/mol), n the valency of the ion.
• Combined law: m = MIt/(nF).

The Faraday constant is the charge carried by one mole of singly charged ions: F = eN_A.

Uses: electroplating (coating an object with a thin metal layer), electroforming (making metal copies), refining metals, producing aluminium, charging batteries.

Worked example

A current of 2 A passes through copper sulphate solution for 1 hour (k = 0.33·10⁻⁶ kg/C). How much copper is deposited on the cathode?

1. Time: t = 3600 s.
2. m = kIt = 0.33·10⁻⁶·2·3600 ≈ 2.4·10⁻³ kg = 2.4 g.
3. Check with the second law: k = 0.064/(2·96 500) ≈ 0.33·10⁻⁶ kg/C — it agrees.

§3. Current in gases

Under ordinary conditions a gas is an insulator: it has almost no free charges. To conduct, it must be ionised: heating, ultraviolet or X-rays, or radioactive radiation knock electrons out of molecules. The carriers in a gas are electrons and positive ions.

• A non-self-sustaining discharge stops when the external ioniser is removed.
• A self-sustaining discharge: in a strong field, electrons gain so much energy between collisions that they ionise molecules by impact themselves, and the current continues without an external ioniser.

Types of self-sustaining discharge:

1. Spark — a brief, powerful discharge (lightning).
2. Arc — a large current and high temperature (welding, searchlights).
3. Corona — a glow around sharp points (high-voltage lines).
4. Glow — at low pressure (gas-discharge lamps, neon signs).

A strongly ionised gas is called plasma — the fourth state of matter; the matter of the Sun and stars is plasma.

Worked example

Suppose an electron travels on average λ = 5·10⁻⁶ m between collisions and 15 eV is needed to ionise a molecule. In what field does impact ionisation begin?

1. Over this distance the electron gains energy eEλ; the condition is eEλ ≥ 15 eV.
2. E ≥ 15/(5·10⁻⁶) = 3·10⁶ V/m.

§4. Current in a vacuum

A vacuum has no carriers, so they must be extracted from an electrode. A heated cathode emits electrons — thermionic emission: some electrons have enough thermal energy to overcome the metal’s work function. A vacuum diode based on this conducts in one direction only (electrons from cathode to anode): if the anode is negative, the electrons do not reach it. In cathode-ray and X-ray tubes the electrons are accelerated by a field:

where e = 1.6·10⁻¹⁹ C and m = 9.1·10⁻³¹ kg. An electron beam can be deflected by an electric or magnetic field.

Worked example

An electron starts from rest and is accelerated through 100 V. What is its speed?

1. v = √(2eU/m) = √(2·1.6·10⁻¹⁹·100/9.1·10⁻³¹).
2. v = √(3.52·10¹³) ≈ 5.9·10⁶ m/s.

§5. Current in semiconductors

In silicon (Si), germanium (Ge) and similar substances there are few free charges at low temperature. Heating or illumination breaks covalent bonds, creating an electron and an empty place — a hole (positively charged). So a semiconductor’s resistance decreases as temperature rises (thermistors and photoresistors are based on this).

• Intrinsic conduction — in a pure crystal the numbers of electrons and holes are equal.
• A donor impurity (pentavalent: P, As) supplies an extra electron → n-type.
• An acceptor impurity (trivalent: B, In) creates a hole → p-type.

A p–n junction: with forward bias (“+” on p) current flows easily, with reverse bias hardly at all. This is a diode, used to rectify alternating current. A transistor, with two p–n junctions, amplifies signals and acts as a switch.

§6. Overall comparison

• Metal: free electrons; resistance rises on heating; no matter is transported.
• Electrolyte: positive and negative ions; resistance falls on heating; substances are released at the electrodes.
• Gas: electrons and ions; the current depends on ionisation, and Ohm’s law usually does not hold.
• Vacuum: electrons produced by emission; current in one direction.
• Semiconductor: electrons and holes; resistance falls on heating or illumination.

Worked example

A resistance thermometer reads 50 Ω at 0 °C, α = 4·10⁻³ K⁻¹. What is the temperature when its resistance is 60 Ω?

1. R = R₀(1 + αt) → t = (R − R₀)/(R₀α).
2. t = (60 − 50)/(50·4·10⁻³) = 50 °C.

Check yourself

1. A current of 5 A passes through copper sulphate for 30 min (k = 0.33·10⁻⁶ kg/C). What mass of copper is deposited?

2. A current of 1 A passes through a silver solution (M = 0.108 kg/mol, n = 1) for 965 s. How much silver is deposited?

3. How long does it take to deposit 6.6 g of copper with a current of 2 A?

4. Arsenic (As, pentavalent) is added to silicon. What type of conduction results?

5. An electron is accelerated through 400 V. What is its kinetic energy in joules?

6. A metal and a semiconductor are heated. How do their resistances change?

7. In electrolysis, to which electrode do positive ions move?

8. What is a discharge that continues without an external ioniser called?

Common mistakes

• Thinking a semiconductor’s resistance rises on heating, like a metal’s.
• Not converting time to seconds and mass from grams to kilograms in Faraday’s law.
• Treating a hole as a separate particle: it is a place where an electron is missing, a positively charged quasiparticle.
• Mixing up donors and acceptors: a pentavalent impurity gives n-type, a trivalent one p-type.
• Thinking cations move to the anode: positive ions go to the cathode (the negative electrode).
• Believing electrons in a wire move at the speed of light: the field travels at that speed, while the electrons’ drift velocity is very small.