Electric Current in Conductors

Until now charges sat still. Let them move and you get an electric current: the thing that lights bulbs, turns motors and runs every circuit.

Electric current#

Whenever charge passes from one side of a surface to the other, there is a current through that surface. If charge Q crosses in time Δt, the average current is Q/Δt; the current at an instant is the rate of flow:

The SI unit is the ampere (A): one coulomb per second. It is one of the seven SI base units. The direction of a current is the direction in which positive charge moves; if the carriers are negative, the current points opposite to their motion.

Current density#

To describe how current is spread over a cross-section we use a vector, the current density.

If the area is tilted so that its normal makes an angle θ with the flow, only the perpendicular part counts: Δi = jΔS cos θ = j⃗·ΔS⃗. For a finite surface:

Where currents come from#

• A hot metal emits electrons that fly through space: a current through any area they cross.
• In a solution, positive ions drift along a field and negative ions against it; both add to the current along the field.
• While a capacitor charges, charge flows along the connecting wires.
• This chapter: free electrons in a metal, pushed by a field inside the conductor.

Electrons inside a metal rush about at hundreds of kilometres a second, yet carry a current by creeping along at a fraction of a millimetre per second.

Free electrons and the lattice#

A metal contains many loosely bound free (conduction) electrons moving among heavy positive ions, the lattice, which vibrate about fixed positions (more strongly when hotter). Every so often an electron collides with the lattice and sets off in a new random direction with a new random speed, so it follows a zig-zag path. With so many electrons moving at random, as many cross any area one way as the other: no current.

Drift in a field#

Apply a field E. Between collisions each electron feels a force eE against the field and picks up extra velocity in that direction, growing steadily until the next collision wipes it out. The random part averages to nothing, but the extra part doesn’t: the electrons slowly drift opposite to the field. If the time between collisions is τ, the extra distance gained per interval is

For a given material at a given temperature τ is fixed, so we can picture the electrons as at rest with no field and moving at a steady drift velocity proportional to E when there is one:

Current and drift speed#

Take a wire of cross-section A with n free electrons per unit volume. In time Δt the electrons in a length v_dΔt, i.e. nAv_dΔt of them, cross a given section, carrying charge nAv_dΔt · e. Dividing by Δt:

Combine the drift picture with i = nAev_d and one of the most used rules in physics falls out: current density is proportional to the field.

Ohm’s law#

Putting v_d = (eτ/2m)E into j = nev_d:

The content of Ohm’s law is that σ does not depend on the field: double E and j doubles. This holds for metals over a wide range of fields.

Resistance#

Take a conductor of length l and uniform cross-section A with potential difference V across its ends. Inside, E = V/l and j = i/A, so i/A = σV/l, i.e. V = (ρl/A) i:

• R is the resistance, in ohms (Ω); 1/R is the conductance.
• Resistivity is in Ω m; conductivity in (Ω m)⁻¹, also written mho m⁻¹.
• A component made to have a chosen resistance is a resistor, drawn as a zig-zag.

The colour code#

Carbon resistors are cylinders with wire leads, labelled by four coloured bands. Read from the end the bands are bunched toward: the first two are digits, the third is the power of ten, the fourth the tolerance (gold ±5 %, silver ±10 %; no fourth band means ±20 %).

Warm a metal and it resists more. Warm a semiconductor and it resists less. Cool certain materials far enough and resistance vanishes altogether.

Metals get worse when hot#

Heating makes the lattice vibrate harder, so electrons collide more often. τ falls, the drift speed falls, and the resistivity rises. Over modest temperature ranges the rise is close to linear:

The length and area of a wire barely change with temperature, so the resistance follows the same rule: R(T) = R(T₀)[1 + α(T − T₀)].

Resistivities span more than 25 powers of ten: metals around 10⁻⁸ Ω m are good conductors, materials like fused quartz are insulators, and silicon and germanium sit in between: semiconductors.

Thermistors#

In semiconductors α is negative and often large: warming them frees more charge carriers, and the resistivity drops. (The simple free-electron picture doesn’t apply here.) A small bead of semiconducting metal oxide (of nickel, iron, cobalt, copper…) sealed in epoxy is a thermistor. Put it in a circuit with a cell: as the temperature rises its resistance falls and the current climbs. Thermistors can detect changes of about 10⁻³ °C.

Superconductors#

Some materials lose their resistance completely below a critical temperature. Kamerlingh Onnes found this in mercury in 1911 (T_c = 4.2 K). Above T_c the material behaves like a normal metal. A current started in a superconducting loop keeps flowing for years with no emf.

A battery is a charge pump. Its emf measures how much work it does on each coulomb, and a resistor turns that work into heat.

How a battery keeps its voltage#

Inside a battery some non-electrostatic mechanism (chemistry, in a cell) pushes positive charge toward terminal A and negative toward B with a force F⃗_b. As charge piles up, an electrostatic field builds between the terminals and pushes back with F⃗_e = qE⃗. Accumulation stops when F_e = F_b.

With nothing connected, F_bd = qEd = qV, so the emf equals the terminal voltage of an open battery. The two are different quantities: emf comes from the non-electrostatic battery force, potential difference from the electrostatic field of the piled-up charges.

Connect a wire across the terminals and charge flows round; the battery mechanism keeps restoring the charge on A and B, maintaining a steady current. Normally positive charge leaves from the + terminal (discharging); driving current into the + terminal instead is charging.

Heat in a resistor#

A resistor R joined to a battery by wires of negligible resistance has potential difference V = iR across it. In time t a charge q = it passes through and loses potential energy qV, which turns into thermal energy:

Internal resistance#

As charge q = it passes through a battery, the battery force does work qℰ, but the charge gains only qV of electric potential energy. The difference warms the battery. An ideal battery wastes nothing, so V = ℰ always. A real battery behaves like an ideal emf in series with an internal resistance r:

Two simple rules, one about charge and one about energy, are enough to solve any circuit of cells and resistors.

The junction law#

The total current flowing into any point of a circuit equals the total current flowing out. Equivalently, counting currents toward the point as positive and away as negative, their algebraic sum is zero.

Why: no point in a circuit keeps piling up charge or keeps producing it, so whatever arrives must leave. It is conservation of charge.

The loop law#

Going all the way round any closed loop, the potential differences add up to zero. Walk round the loop in one direction and list each change: count a drop in potential as positive and a rise as negative (or the other way round, consistently).

• Through a resistor in the direction of its current: drop iR. Against the current: rise iR.
• Through a cell from − to +: rise ℰ. From + to −: drop ℰ.

Why: the electrostatic force is conservative, so its work round a closed path, and hence the net change of potential, is zero. You end where you started.

Any two-terminal network of resistors acts like one resistor. Series and parallel rules handle most of them; the same ideas tell us how to group cells.

Equivalent resistance#

Apply potential difference V across a network’s two end points; if it draws current i, its equivalent resistance is R_eq = V/i.

Series#

Resistors are in series when the same current passes through each. Round the loop: iR₁ + iR₂ + iR₃ − V = 0, so

Parallel#

Resistors are in parallel when the same potential difference is across each. The current splits: i = i₁ + i₂ + i₃ = V/R₁ + V/R₂ + V/R₃, so

How current divides#

Across two parallel resistors V is common, so i₁R₁ = i₂R₂: the current divides in inverse ratio of the resistances.

Grouping cells#

Series: two cells (ℰ₁, r₁) and (ℰ₂, r₂) driving R. Round the loop, i = (ℰ₁ + ℰ₂)/(R + r₁ + r₂): they act as one cell of emf ℰ₁ + ℰ₂ and internal resistance r₁ + r₂. Reverse one cell and the emf becomes |ℰ₁ − ℰ₂|.

Parallel: applying the loop law to each cell’s loop through R and eliminating gives one equivalent cell:

Measuring electricity means putting an instrument into the circuit. The Wheatstone bridge measures resistance by finding a null; ammeters and voltmeters are designed to disturb the circuit as little as possible.

The Wheatstone bridge#

Four resistors R₁, R₂, R₃, R₄ form a loop with corners A, C, B, D. A cell is joined across A and B, and a galvanometer (a sensitive meter whose needle swings left or right with the current direction) across C and D.

For one particular set of values no current flows through the galvanometer: the bridge is balanced. Then V_C = V_D, R₁ and R₂ carry the same current, and so do R₃ and R₄. Comparing the drops, (V_A − V_C)/(V_C − V_B) = R₁/R₂ and (V_A − V_D)/(V_D − V_B) = R₃/R₄. With V_C = V_D these ratios are equal:

To measure an unknown R₄, make R₃ variable and adjust it until the galvanometer reads zero; then R₄ = R₂R₃/R₁.

Ammeters and voltmeters#

Both contain a coil hung between magnet poles; current through the coil turns it, and a needle fixed to it moves along a scale by an amount proportional to the current.

Ammeter

It goes in series with the part whose current you want, current entering at the + terminal. A small resistance r, the shunt, is joined in parallel with the coil (resistance R_c), so the coil takes only a fixed fraction of the current, i₁ = ri/(R_c + r). Its total resistance is

A galvanometer is built the same way but with its zero in the middle of the scale, so it shows current in either direction.

Voltmeter

It goes in parallel with the part whose voltage you want, + terminal to the higher potential. A large resistance R is joined in series with the coil, so the coil current is V/(R_c + R), proportional to V. Its resistance R_c + R must be large, so that it draws little current and barely changes the voltage it measures.

An ideal voltmeter would draw no current at all. A potentiometer achieves exactly that by balancing the unknown voltage against a known one until nothing flows.

The stretched-wire potentiometer#

A long uniform wire AB (5–10 m, laid in 1 m lengths joined by thick copper strips) is fixed to a board beside a metre scale. A driver circuit, a strong cell with a key and a rheostat, sends a steady current through it, so the potential falls evenly from A to B. A galvanometer is connected to a jockey that can touch the wire at any point P.

To measure the potential difference between points a and b (a higher), join a to A and b, through the galvanometer, to the jockey. Slide the jockey until the galvanometer shows no deflection. At that point P has the same potential as b, so V_ab = V_AP. Because the wire is uniform, the drop is proportional to length: with V₀ across the whole length L,

Calibration#

To know V₀, balance a standard cell of accurately known emf ℰ: if it balances at l₀, then ℰ = V₀l₀/L, so V₀ = Lℰ/l₀ and any later reading is V = (l/l₀)ℰ. (No current flows through the standard cell at balance, so its terminal voltage is its emf.)

Uses#

Comparing two emfs

Balance the first cell at l₁, then the second at l₂ (a two-way key swaps them). Since ℰ₁ = V₀l₁/L and ℰ₂ = V₀l₂/L,

Internal resistance of a cell

Connect a known resistance R across the cell through a key K. With K open, no current flows from the cell, so it balances at l with ℰ = V₀l/L. Close K: current ℰ/(R + r) flows and the terminal voltage drops to ℰR/(R + r), balancing at a shorter l′. Dividing the two:

A capacitor joined to a battery through a resistor doesn’t fill instantly. The charge creeps up along an exponential curve whose pace is set by one number, CR.

Charging#

A capacitor C, a cell ℰ and a resistance R (the wires plus the cell’s internal resistance) are joined in series at t = 0. At time t the capacitor holds q and the current is i = dq/dt. Round the loop:

The product CR has the dimensions of time; it is the time constant τ. The final charge ℰC is reached only as t → ∞, but after one time constant q = ℰC(1 − 1/e) = 0.63 ℰC.

Discharging#

A capacitor holding Q is joined across a resistance R at t = 0. Now the charge falls, so the current is i = −dq/dt, and the loop law gives q/C − Ri = 0, i.e. R dq/dt = −q/C:

After one time constant 37 % (1/e) of the charge is left, so 63 % of the discharge is done. Again it is complete only at t → ∞, but four or five time constants leave less than 2 %.

The Earth and its atmosphere form a giant leaky capacitor, charged by thunderstorms and discharged by fair weather everywhere else.

The fair-weather field#

The ground carries a negative charge, a little under 1 nC per square metre, spread fairly evenly. Above it there is a downward electric field of about 100 V/m: over a flat desert, the potential rises by roughly 100 V for every metre you climb. Higher up the field weakens, becoming negligible around 50 km, where the potential is about 400 kV above the ground.

A current in the air#

Cosmic rays ionise air molecules, and dust picks up charge by friction, so the air holds both positive and negative ions. The field pushes positive ions down and negative ions up: a vertical current of about 3.5 × 10⁻¹² A through every square metre. The density of ions grows with height and thinner air lets them travel further between collisions, so the air conducts better higher up; by about 50 km it is highly conducting. Picture a conducting shell at +400 kV, 50 km up, leaking current down to the Earth.

Thunderstorms recharge the battery#

Rising and sinking air carries water vapour into the cold upper atmosphere (about −10 °C at 3–4 km, −20 °C at 6–7 km), where it forms droplets and ice. A mature thunderstorm is roughly 7–8 km across, with its base 3–4 km up and its top at 6–7 km. Its top holds excess positive charge and its base a dense negative charge, putting the cloud base 20–100 MV below the ground’s potential: far larger than, and opposite to, the 400 kV fair-weather difference.

The field below the cloud breaks down the air, and negative charge leaps to the ground: lightning. One stroke delivers about 20 C, and a cloud can recharge for the next in about 5 s. Meanwhile the storm’s positive top leaks up to the conducting layer. Thousands of storms a day thus pump negative charge into the Earth and positive charge upward, acting as the emf; the fair-weather air is the load that discharges it.