Lesson 12 of 12 · 18 min
Chapter review
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Must-know facts
18 facts
- 01I = dq/dt; ampere is an SI base unit; current is a scalar.
- 02Conventional current flows opposite to electron motion.
- 03V = IR; R = ρl/A; ρ in Ω m depends on material and temperature only.
- 04j = σE with σ = 1/ρ; unit of j is A m⁻².
- 05Drift velocity v_d = −eEτ/m; I = neAv_d.
- 06ρ = m/(ne²τ); mobility μ = |v_d|/E = eτ/m, unit m² V⁻¹ s⁻¹.
- 07Drift speed is tiny (about mm/s or less); signals travel fast because the field is set up almost instantly.
- 08In a tapering wire with steady current, current is the same everywhere; drift speed and current density are larger at the narrow end.
- 09Stretching 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
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.