Lesson 12 of 12 · 18 min

Chapter review

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Must-know facts

18 facts

1I = dq/dt; ampere is an SI base unit; current is a scalar.
2Conventional current flows opposite to electron motion.
3V = IR; R = ρl/A; ρ in Ω m depends on material and temperature only.
4j = σE with σ = 1/ρ; unit of j is A m⁻².
5Drift velocity v_d = −eEτ/m; I = neAv_d.
6ρ = m/(ne²τ); mobility μ = |v_d|/E = eτ/m, unit m² V⁻¹ s⁻¹.
7Drift speed is tiny (about mm/s or less); signals travel fast because the field is set up almost instantly.
8In a tapering wire with steady current, current is the same everywhere; drift speed and current density are larger at the narrow end.
9Stretching a wire to n times its length at constant volume makes R n² times.
10Metals: α > 0 because τ decreases on heating; semiconductors: α < 0 because n increases on heating.
11Nichrome, manganin and constantan: high resistivity, low temperature coefficient.
12P = VI = I²R = V²/R; rated resistance R = V²/P.
13In series the lower-wattage bulb glows brighter; in parallel the higher-wattage bulb glows brighter.
14Terminal voltage V = ε − Ir while discharging; V = ε on open circuit.
15Series cells: emfs and internal resistances add (reversed cell's emf subtracts).
16n identical cells in parallel: emf ε, internal resistance r/n.
17Junction rule = charge conservation; loop rule = energy conservation.
18Balanced Wheatstone bridge: P/Q = R/S (P, Q along one path; R, S along the other, each listed from the same battery corner), i.e. products of opposite arms are equal; the galvanometer arm carries no current.

Common traps

Where marks are lost

Converting 2.5 mm² to 2.5 × 10⁻³ m² in drift-speed or resistance questions.

1 mm = 10⁻³ m, so 1 mm² = 10⁻⁶ m². Square the conversion factor for areas.

Thinking that stretching a wire to 1.25 times its length raises its resistance by 25%.

Volume is fixed, so area falls in the same ratio as length rises: R ∝ l², giving (1.25)² = 1.5625 times, a 56.25% rise.

Believing the current or drift speed is the same at both ends of a tapering wire, or that current is larger at the thick end.

Steady current is the same at every cross-section; since I = neAv_d, v_d is larger where A is smaller.

Explaining the rise of resistance of metals with temperature by a change in the number of free electrons.

In metals n is nearly constant; it is the relaxation time τ that decreases as collisions become more frequent.

Assuming a 100 W bulb always glows brighter than a 25 W bulb.

In series the same current flows, so power ∝ R. The 25 W bulb has four times the resistance and dissipates more power in series.

Treating the emf of a cell as the voltage across its terminals while it supplies current.

Terminal voltage is ε − Ir during discharge; it equals ε only when no current is drawn.

Mixing up sign rules in Kirchhoff's loop equations, especially across cells.

Pick a direction to go round the loop; resistor: −IR along the assumed current, +IR against it; cell: +ε going from − to +, −ε going from + to −.

Assuming a five-resistor bridge network is balanced and dropping the middle resistor without checking.

Check the ratio of the two resistances in each path. Only if they match is the middle arm current-free; otherwise solve with Kirchhoff's rules.

Using the series formula for internal resistance when two cells are in parallel.

For parallel cells r_eq = r₁r₂/(r₁ + r₂) and ε_eq = (ε₁r₂ + ε₂r₁)/(r₁ + r₂).

Formulas

19 to know

Electric current

I = dq/dt (steady: I = q/t)

SI unit ampere (A).

Current density

j = I/A

A is the area normal to the flow; unit A m⁻².

Ohm's law

V = IR

R in ohm (Ω).

Resistance of a uniform conductor

R = ρl/A

ρ in Ω m.

Ohm's law, vector form

j = σE, σ = 1/ρ

σ in S m⁻¹ (Ω⁻¹ m⁻¹).

Drift velocity

v_d = −eEτ/m

τ is the relaxation time; negative sign: electrons drift opposite to E.

Current and drift speed

I = neAv_d

n = free electrons per unit volume.

Resistivity from microscopic quantities

ρ = m/(ne²τ)

Explains the temperature dependence of ρ.

Mobility

μ = |v_d|/E = eτ/m

Unit m² V⁻¹ s⁻¹.

Temperature dependence of resistivity

ρ_T = ρ₀[1 + α(T − T₀)]

α > 0 for metals; α < 0 for semiconductors.

Electrical power

P = VI = I²R = V²/R

Heat dissipated in a resistor.

Power loss in transmission

P_c = P²R_c/V²

P delivered at voltage V through cables of resistance R_c.

Cell with internal resistance

I = ε/(R + r); V = ε − Ir

V is the terminal voltage while the cell supplies current.

Cells in series

ε_eq = ε₁ + ε₂; r_eq = r₁ + r₂

Subtract an emf for a reversed cell.

Cells in parallel

ε_eq = (ε₁r₂ + ε₂r₁)/(r₁ + r₂); r_eq = r₁r₂/(r₁ + r₂)

Like terminals joined; change the sign of ε for a reversed cell.

Kirchhoff's junction rule

ΣI_in = ΣI_out

Conservation of charge.

Kirchhoff's loop rule

Σ(changes in potential) = 0 round any closed loop

Conservation of energy.

Wheatstone bridge balance

P/Q = R/S (equivalently PS = QR)

Battery across A and C, galvanometer across B and D; P = AB, Q = BC on one path, R = AD, S = DC on the other. Labels are ours; with any labelling, the resistors meeting at a battery corner are in the same ratio as the ones following them on their paths.

Resistors in series and parallel

R_s = R₁ + R₂ + …; 1/R_p = 1/R₁ + 1/R₂ + …

Background from earlier classes, used throughout network problems.

Key terms

15 terms

Electric current: The net rate of flow of charge through a cross-section.
Current density: Current per unit area, with the area taken normal to the direction of flow.
Resistivity: A material's inherent opposition to current, independent of the sample's dimensions.
Conductivity: The reciprocal of resistivity.
Relaxation time: The average time between successive collisions of a free electron.
Drift velocity: The small average velocity of free electrons along a conductor due to an applied field.
Mobility: Drift speed of a charge carrier per unit electric field.
Non-ohmic device: A material or device whose current does not vary in direct proportion to voltage.
Temperature coefficient of resistivity: The fractional change in resistivity per unit change in temperature.
Joule heating: Conversion of electrical energy into heat in a resistor, at a rate I²R.
emf: The potential difference between the terminals of a cell when it supplies no current.
Internal resistance: The resistance to current inside a cell, mainly from its electrolyte.
Terminal voltage: The potential difference across a cell's terminals when it is in a circuit.
Junction: A point in a circuit where three or more conductors meet.
Wheatstone bridge: A four-resistor arrangement used to compare resistances by finding a null in a galvanometer.

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