1Electric Current

Electric current is the rate at which charge flows past a point in a conductor.

Definition
I = dq / dt
Unit: ampere (A), where 1 A = 1 C/s. By convention, current direction is taken as the direction positive charge would flow — opposite to the actual motion of electrons in a metal.

2Drift Velocity and the Origin of Resistivity

Free electrons in a conductor are already moving fast in random directions due to thermal motion — but with no net displacement, so no current flows without a field. Apply an electric field, and each electron picks up a small extra velocity between collisions with the lattice, giving a slow net drift superimposed on the random motion.

Current in Terms of Drift Velocity
I = n A e v_d
n = free electron density, A = cross-sectional area, e = electronic charge, v_d = drift velocity. Drift velocities in typical wires are surprisingly slow — often less than a millimetre per second — even though the current “signal” itself travels near the speed of light.

Between collisions, an electron accelerates freely under the field for an average time τ (relaxation time), giving:

Drift Velocity
v_d = eEτ / m
Combining this with I = nAev_d and Ohm’s Law leads to the microscopic definition of resistivity: ρ = m/(ne²τ) — resistivity comes from how often and how effectively electrons collide with the lattice.

3Ohm’s Law

For many conductors at constant temperature, the current through them is directly proportional to the potential difference applied across them.

Ohm’s Law
V = I R
R (resistance) is the constant of proportionality, in ohms (Ω). It depends on the material and geometry of the conductor: R = ρL/A, where ρ is resistivity, L is length, A is cross-sectional area.

4Limitations of Ohm’s Law

Not every conducting device obeys Ohm’s Law — such devices are called non-ohmic. Common exceptions worth remembering:

• Diodes: current depends on voltage in a strongly non-linear way, and depends on polarity (conducts in one direction, blocks the other).
• Filament lamps: V–I graph curves, since resistance rises as the filament heats up.
• Electrolytes and gas discharge tubes: V–I relationship depends on additional factors beyond just voltage.

Exam Tip
A V–I graph that’s a straight line through the origin means the device is ohmic. Any curve, kink, or offset means it isn’t.

5Temperature Dependence of Resistivity

Resistivity vs. Temperature
ρ(T) = ρ₀ [1 + α(T − T₀)]
α is the temperature coefficient of resistivity. For most metals, α is positive — resistivity rises with temperature, since increased thermal vibration means more frequent electron collisions (shorter τ). For semiconductors, α is negative: resistivity falls as temperature rises, since more charge carriers become available.

6Electrical Energy and Power

Pushing current through a resistor dissipates energy as heat — the same effect used deliberately in heaters, and unavoidably in every real circuit.

Power Dissipated (three equivalent forms)
P = V I = I² R = V² / R
Use whichever form matches the quantities you’re given — all three are algebraically identical via V = IR.

7Resistors in Series and Parallel

Series Combination

Equivalent Resistance
R = R₁ + R₂ + R₃ + …
Same current flows through each resistor; voltages add up. Equivalent resistance is always larger than the largest individual resistor.

Parallel Combination

Equivalent Resistance
1/R = 1/R₁ + 1/R₂ + 1/R₃ + …
Same voltage across each resistor; currents add up. Equivalent resistance is always smaller than the smallest individual resistor.

8Cells, EMF and Internal Resistance

A real cell isn’t a perfect voltage source — it has internal resistance r that dissipates some energy inside the cell itself, so the voltage delivered to an external circuit is always a little less than the cell’s full EMF.

Circuit Current
ε = I(R + r)
ε = EMF, R = external resistance, r = internal resistance.
Terminal Voltage
V = ε − Ir
Terminal voltage drops as current increases — the more current you draw, the more voltage is “lost” across the internal resistance.

Cells in Series and Parallel

• n identical cells in series: total EMF = nε, total internal resistance = nr — used when you need higher voltage.
• n identical cells in parallel: total EMF = ε, total internal resistance = r/n — used when you need higher current capacity without raising voltage.

9Kirchhoff’s Laws

For circuits too complex to reduce with simple series/parallel rules, Kirchhoff’s two laws provide a systematic way to solve for every current and voltage.

Junction Rule (Kirchhoff’s Current Law)

At any junction
Σ I_in = Σ I_out
A direct statement of charge conservation — charge can’t pile up or vanish at a junction.

Loop Rule (Kirchhoff’s Voltage Law)

Around any closed loop
Σ ΔV = 0
The sum of all potential changes (EMF gains and IR drops) around any closed loop is zero — a statement of energy conservation.

10Wheatstone Bridge

A Wheatstone bridge is four resistors arranged in a diamond, with a galvanometer connecting the two midpoints, used to find an unknown resistance precisely by balancing against known ones.

Balance Condition
P / Q = R / S
When the bridge is balanced (galvanometer reads zero), no current flows through the galvanometer branch, and the ratio of resistances on each arm are equal. A meter bridge is a practical, laboratory version of the same idea, using a uniform wire in place of two of the resistors.

11Potentiometer

A potentiometer compares potential differences by balancing them against the potential drop along a uniform wire carrying a steady current — without drawing any current from the source being measured, which makes it more accurate than a voltmeter for this purpose.

Comparing Two EMFs
ε₁ / ε₂ = l₁ / l₂
l₁ and l₂ are the balancing lengths on the potentiometer wire for each EMF. Since no current is drawn at balance, this method measures true EMF, not just terminal voltage — also used to measure a cell’s internal resistance.

Formula Summary

CurrentI = dq/dt
Drift VelocityI = nAev_d
Ohm’s LawV = IR
ResistanceR = ρL/A
Resistivity vs Tρ = ρ₀[1+α(T−T₀)]
PowerP = VI = I²R
Series RR = ΣRᵢ
Parallel R1/R = Σ1/Rᵢ
Terminal VoltageV = ε − Ir
Wheatstone BridgeP/Q = R/S
Potentiometerε₁/ε₂ = l₁/l₂

Solved Examples

Example 1 · Drift Velocity

A copper wire of cross-sectional area 1×10⁻⁶ m² carries a current of 1.5 A. Given free electron density n = 8.5×10²⁸ m⁻³, find the drift velocity.

Solution: v_d = I/(nAe) = 1.5 / (8.5×10²⁸ × 1×10⁻⁶ × 1.6×10⁻¹⁹) = 1.5 / (1.36×10⁴) ≈ 1.1×10⁻⁴ m/s.

Notice how slow that is — barely a tenth of a millimetre per second, despite the “instant” feel of switching on a light.

Example 2 · Cells and Internal Resistance

A cell of EMF 2 V and internal resistance 0.5 Ω is connected to an external resistance of 4.5 Ω. Find the current in the circuit and the terminal voltage.

Solution: I = ε/(R+r) = 2/(4.5+0.5) = 2/5 = 0.4 A.

Terminal voltage: V = ε − Ir = 2 − (0.4×0.5) = 2 − 0.2 = 1.8 V.

Quick Check

1. Two resistors of 2 Ω and 3 Ω are connected in parallel. The equivalent resistance is:
✓ 1.2 Ω (correct)
5 Ω
6 Ω
0.5 Ω
2. Kirchhoff’s junction rule is a direct consequence of the conservation of:
✓ Charge (correct)
Energy
Momentum
Mass
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