Current, Resistance and Resistivity

Current and charge

Current (I) is the rate of flow of charge:

I = ΔQ ÷ Δt      (amps = coulombs per second)

Current is carried by moving charges (electrons in metals). The relationship to charge-carrier movement:

I = nAvq

where n = number density of charge carriers, A = cross-sectional area, v = drift velocity, q = charge on each carrier.

Potential difference and resistance

• Potential difference (V) = energy transferred per unit charge (volts = J/C).
• Resistance (R) = V ÷ I (ohms). Ohm's law: for an ohmic conductor at constant temperature, V ∝ I (constant R).

I–V characteristics

• Ohmic conductor / resistor: straight line through the origin.
• Filament lamp: curve — resistance increases as it heats up.
• Diode: conducts one way only; very high resistance in reverse.

Resistivity

Resistance depends on the material and dimensions:

R = ρL ÷ A

where ρ = resistivity (Ω m) — a property of the material, L = length, A = cross-sectional area. Longer/thinner wires have more resistance. Resistivity increases with temperature for metals.

Superconductivity

Below a critical temperature, some materials have zero resistivity (superconductors) — no energy is dissipated, useful for very strong electromagnets and lossless power transmission.

Power and energy in circuits

P = VI = I²R = V² ÷ R
E = VIt = Pt

Series and parallel resistors

• Series: R_total = R₁ + R₂ + … (current the same, p.d. shared).
• Parallel: 1/R_total = 1/R₁ + 1/R₂ + … (p.d. the same, current shared); total resistance is less than the smallest resistor.

Worked example

A wire has resistivity 1.7 × 10⁻⁸ Ω m, length 2.0 m and cross-sectional area 1.0 × 10⁻⁶ m². Find its resistance.

• R = ρL/A = (1.7×10⁻⁸ × 2.0) ÷ 1.0×10⁻⁶ = 3.4×10⁻⁸ ÷ 1.0×10⁻⁶ = 0.034 Ω. ✓

Common mistakes

• Rearranging R = ρL/A incorrectly, or mixing up length and area effects.
• Forgetting a filament lamp's resistance rises with temperature (non-ohmic).
• Adding parallel resistances like series (use the reciprocal rule).

Exam tips

• Learn I = nAvq and be able to explain drift velocity.
• Use R = ρL/A; resistivity is a material property (independent of dimensions).
• Know the three power equations and the series/parallel resistor rules.

Key facts to remember

• I = ΔQ/Δt = nAvq; V = IR (Ohm's law for ohmic conductors at constant temperature).
• Resistivity R = ρL/A (material property); superconductors have zero resistivity below a critical temperature.
• Power P = VI = I²R = V²/R; series resistances add, parallel add as reciprocals.