Electric current and drift velocity
Current is the rate of flow of charge: I = dq/dt, measured in amperes.
In a metal, free electrons move randomly at high speeds, but with no net flow. When a field E is applied, they acquire a small average velocity opposite to E, called the drift velocity v_d:
v_d = −eEτ / m
where τ is the average time between collisions (the relaxation time). Drift velocity is tiny, typically around 10⁻⁴ m/s. The current is
I = n e A v_d
where n is the number density of free electrons and A the cross-sectional area.
A bulb lights almost instantly when switched on, despite the tiny drift velocity, because the electric field is established throughout the wire almost at the speed of light, and all the electrons start drifting at once.
Ohm's law and resistivity
V = I R, and R = ρ l / A
The resistivity ρ is a property of the material: ρ = m / (n e² τ). In microscopic form, Ohm's law reads J = σE, where J = I/A is the current density and σ = 1/ρ the conductivity.
Mobility μ = v_d / E is the drift velocity per unit field.
Temperature dependence
Metals: resistivity increases with temperature, because ions vibrate more and τ decreases. Approximately ρ_T = ρ₀[1 + α(T − T₀)], where α is the temperature coefficient of resistivity.
Semiconductors: resistivity decreases with temperature, because more charge carriers become available.
Alloys such as manganin and constantan have very small α, so they are used for standard resistors.
Ohm's law is not universal. Diodes, for example, are non-ohmic: their V–I graph is not a straight line.
Electrical power
P = V I = I² R = V² / R
EMF and internal resistance
The emf ε of a cell is the potential difference between its terminals when no current is drawn. A real cell has internal resistance r. When it drives a current I through an external resistance R:
I = ε / (R + r), and terminal voltage V = ε − I r
So the terminal voltage is less than the emf while the cell is supplying current.
Example. A 12 V battery with internal resistance 0.5 Ω is connected to a 5.5 Ω resistor. I = 12 / 6 = 2 A, and the terminal voltage is 12 − 2 × 0.5 = 11 V.
Cells in combination
Series: ε_eq = ε₁ + ε₂ and r_eq = r₁ + r₂.
Parallel: ε_eq = (ε₁r₂ + ε₂r₁)/(r₁ + r₂) and 1/r_eq = 1/r₁ + 1/r₂.
Kirchhoff's rules
Junction rule (conservation of charge): the sum of the currents entering a junction equals the sum of the currents leaving it.
Loop rule (conservation of energy): the algebraic sum of the changes in potential around any closed loop is zero.
Sign convention for the loop rule: going through a resistor in the direction of current gives a potential drop of −IR. Going from the negative to the positive terminal of a cell gives a rise of +ε.
Wheatstone bridge
Four resistors P, Q, R, S are arranged in a diamond, with a galvanometer across the middle. The bridge is balanced, with no current through the galvanometer, when
P / Q = R / S
The metre bridge uses this principle to measure an unknown resistance with a uniform 1 m wire. At the balance point at length l (in cm): R / S = l / (100 − l).