Electric Current & Drift Velocity
Electric current is the rate of flow of charge through any cross-section of a conductor.
SI unit ampere: 1 A = 1 C s⁻¹. Current is a scalar — it has a direction but adds by ordinary (scalar) addition. The charge that flows is the area under the I–t graph.
direction of flow of positive charge (+ to − outside the cell)
direction of flow of electrons — opposite to the conventional current
Who carries the current?
| Medium | Charge carriers |
|---|---|
| Metals (conductors) | free electrons |
| Semiconductors | free electrons and holes |
| Liquids (electrolytes) | positive and negative ions |
| Gases (ionised) | positive ions and electrons |
Slow average speed of free electrons
Drift velocity vd is the average velocity with which free electrons drift towards the positive end under an applied field.
τ = relaxation time (mean time between collisions, ~10⁻¹⁴ s); n = free electrons per m³. vd is only ~mm/s. For a steady current in a wire of changing area, I is the same everywhere, so vd ∝ 1/A. As temperature rises, τ falls, so vd falls.
Mobility & Current Density
Mobility is the drift velocity per unit electric field.
SI unit m² V⁻¹ s⁻¹. Electrons are more mobile than holes (μe > μh) because electrons move freely while holes move from bond to bond.
Current per unit area
Current density J at a point is the current per unit area held normal to the current. J is a vector along the current.
SI unit A m⁻², dimension [L⁻² A]. If the area makes angle θ with the normal to the current: I = JA cosθ.
Linking J, σ and E
Ohm’s Law, Resistance & Resistivity
If the physical state of a conductor (temperature, strain) stays the same, the current through it is proportional to the p.d. across it.
R is the resistance: 1 Ω = 1 V A⁻¹, dimension [M L² T⁻³ A⁻²]. The V–I graph of an ohmic conductor is a straight line through the origin; slope of V–I = R.
metals at constant temperature — straight-line V–I graph
diode, transistor, vacuum tube, electrolyte, discharge tube — curved V–I graph
A property of the material only
ρ (Ω m) does not depend on the shape or size of the body, only on the material and temperature. Conductivity σ = 1/ρ (S m⁻¹); conductance G = 1/R (siemens, mho). Order: ρinsulator > ρalloy > ρsemiconductor > ρconductor.
Volume stays constant
| Wire stretched so that | New resistance | Small % change |
|---|---|---|
| length becomes n times | R′ = n²R | ΔR/R = 2 Δl/l |
| radius becomes 1/n times | R′ = n⁴R | ΔR/R = 4 Δr/r |
| cut into n equal parts, joined in parallel | R′ = R/n² | — |
Temperature Dependence & Colour Code
For metals the resistance rises with temperature because ions vibrate more and electrons collide more often (τ falls).
| Material | α | R on heating |
|---|---|---|
| Metals | positive | increases |
| Alloys (manganin, constantan, nichrome) | very small +ve | almost constant |
| Semiconductors, insulators, electrolytes | negative | decreases |
Alloys with small α and high ρ are used for standard resistors. Superconductors: below a critical temperature the resistance falls to zero (used in MRI magnets, loss-free cables). Thermistors: NTC (R falls on heating) are temperature sensors; PTC (R rises on heating) act as resettable fuses.
Carbon resistors: B B R O Y G B V G W
| Colour | Digit / multiplier | Colour | Digit / multiplier |
|---|---|---|---|
| Black | 0 / 10⁰ | Green | 5 / 10⁵ |
| Brown | 1 / 10¹ | Blue | 6 / 10⁶ |
| Red | 2 / 10² | Violet | 7 / 10⁷ |
| Orange | 3 / 10³ | Grey | 8 / 10⁸ |
| Yellow | 4 / 10⁴ | White | 9 / 10⁹ |
4th band (tolerance): gold ±5%, silver ±10%, no band ±20%. Gold and silver as 3rd band mean ×10⁻¹ and ×10⁻².
Combination of Resistors
In series the same current flows through each resistor; in parallel each resistor has the same p.d.
Rs is larger than the largest resistor; Rp is smaller than the smallest one.
Asked again and again
| Situation | Result |
|---|---|
| n identical resistors R in series / in parallel | nR / R/n (ratio Rs/Rp = n²) |
| Wire R cut into n equal parts, all in parallel | R/n² |
| Wire R bent into a circle: across a diameter | R/4 |
| Arc θ of the circle against the rest | R1R2/(R1 + R2) with R1 = Rθ/2π |
| Wire R bent into a triangle: across any side | 2R/9 |
| Bridge network with P/Q = R/S | remove the middle arm (no current) |
Cells: EMF & Internal Resistance
EMF E is the p.d. across the terminals of a cell when no current is drawn. When current flows, part of E is lost inside the cell.
Internal resistance r depends on the nature and separation of the electrodes, the electrolyte (nature, concentration, temperature) and the area of the electrodes dipped in it.
Charging: V = E + Ir (V > E)
Short circuit: R = 0, I = E/r, V = 0
Cells in series and in parallel
| n identical cells | Eeq | req | Current in R | Best when |
|---|---|---|---|---|
| Series | nE | nr | nE/(R + nr) | R ≫ r |
| Parallel | E | r/n | nE/(nR + r) | R ≪ r |
| Mixed (n in series, m rows) | nE | nr/m | mnE/(mR + nr) | R = nr/m |
One cell reversed in a series row reduces the emf by 2E. Two cells of equal emf in series give zero p.d. across cell 1 when R = r₁ − r₂.
Kirchhoff’s Laws & Measuring Instruments
KCL: the algebraic sum of currents meeting at a junction is zero (current in = current out).
KVL: the algebraic sum of all p.d.s round any closed loop is zero.
Going round a loop
| Element crossed | Change in potential |
|---|---|
| Resistor along the current | − IR (drop) |
| Resistor against the current | + IR (rise) |
| Cell from − to + | + E |
| Cell from + to − | − E |
Galvanometer → ammeter / voltmeter
Ammeter: RA = GS/(G + S), low — ideal 0, joined in series. Voltmeter: RV = G + R, high — ideal ∞, joined in parallel. To raise a voltmeter range from V to nV add (n − 1)RV in series; a larger range means a larger resistance. A real voltmeter draws current, so it reads less than the true p.d.
Wheatstone Bridge & Metre Bridge
A Wheatstone bridge is four resistors with a galvanometer across one diagonal. When the galvanometer shows no deflection, VB = VD and the bridge is balanced.
At balance the cell and galvanometer can be interchanged without disturbing balance. The bridge is most sensitive when all four resistances are of the same order. A balanced bridge’s middle arm can be removed when finding equivalent resistance.
Slide-wire form of the Wheatstone bridge
The 1 m wire is of manganin/constantan (high ρ, low α). If R and S are interchanged, the null point moves from l to 100 − l. Specific resistance of the wire in a gap: ρ = Sπr²/L.
Potentiometer
When a steady current flows through a uniform wire, the p.d. across any length is proportional to that length.
K is the potential gradient (V m⁻¹). At the null point no current is drawn from the cell, so the potentiometer measures the true emf: E = Kl.
Three standard experiments
| Experiment | Result |
|---|---|
| Compare emfs of two cells | E₁/E₂ = l₁/l₂ |
| Sum and difference method | (E₁ + E₂)/(E₁ − E₂) = l₁/l₂ |
| Internal resistance (cell shunted by R) | r = R(l₁/l₂ − 1) |
| Driving cell E₀ with series Rs, wire resistance Rw | K = E₀Rw/[(Rw + Rs + r₀)L] |
Sensitivity is higher when K is smaller: use a longer wire or reduce the current (increase Rh). With a fixed p.d. across a longer wire, K falls and the null point moves further. No balance is possible if the p.d. across the whole wire is less than the emf being measured.
| Voltmeter | Potentiometer |
|---|---|
| High but finite resistance; draws current | Infinite resistance at balance; draws no current |
| Deflection method, reads less than true p.d. | Null method, reads the true emf; not direct reading |
Electrical Power & Heating Effect
Electric power is the rate at which electrical energy is used; the energy appears as heat in a resistor (Joule heating).
1 kWh (1 unit) = 1000 W × 3600 s = 3.6 × 10⁶ J. Units used = (watt × hour)/1000. 1 HP = 746 W.
Same R, different supply
| Series (same I) | Parallel (same V) | |
|---|---|---|
| Total power of bulbs | 1/P = 1/P₁ + 1/P₂ + … | P = P₁ + P₂ + … |
| Brighter bulb | lower wattage (higher R) | higher wattage |
| Two heaters boiling water | t = t₁ + t₂ | t = t₁t₂/(t₁ + t₂) |
| Two identical heaters, same V | Hparallel : Hseries = 4 : 1 | |
Fuse wire: high resistivity and low melting point (alloy of lead and tin), joined in series. Heater element: high resistivity, high melting point (nichrome). Small change: ΔP/P = 2ΔV/V. Chemical effect: in electrolysis the mass deposited m = ZIt (Faraday’s first law).
Just see this before the exam
Whole chapter on one page
I = q/t = dq/dt
vd = eEτ/m
I = neAvd, μ = vd/E
J = I/A = σE
V = IR, R = ρl/A
σ = ne²τ/m = 1/ρ
Stretch: R ∝ l² ∝ 1/r⁴
R = R₀(1 + αΔT)
Rs = R₁ + R₂ + …
1/Rp = 1/R₁ + 1/R₂ + …
I₁ = IR₂/(R₁ + R₂)
Circle, diameter: R/4
I = E/(R + r), V = E − Ir
r = (E/V − 1)R
Series nE/(R + nr)
Pmax at R = r
S = IgG/(I − Ig)
R = V/Ig − G
P/Q = R/S
S = R(100 − l)/l
K = V/L, E = Kl
r = R(l₁/l₂ − 1)
P = VI = I²R = V²/R
1 kWh = 3.6 × 10⁶ J
Avoid these = easy marks
Stretched wire: R ∝ l
Using E for the terminal voltage
Metre bridge: forgetting 100 − l
Bulbs in series: high watt glows more
Volume fixed ⇒ R ∝ l²
V = E − Ir when current flows
R/S = l/(100 − l)
Lower watt (higher R) glows more