Ohm’s law and resistance

Ohm’s law, resistivity and temperature

Ohm’s law: at constant temperature, the current through a conductor is proportional to the potential difference across it: V = IR. R is the resistance, measured in ohm (Ω). A graph of V against I for such an ohmic conductor is a straight line through the origin.

Not all devices obey Ohm’s law. Non-ohmic examples include a semiconductor diode (current depends on the direction of V and is non-linear) and a filament bulb (resistance rises as it heats).

Resistivity: R = ρL/A, where L is length and A is cross-sectional area. ρ is the resistivity, a property of the material only, unit Ω m. Conductivity σ = 1/ρ, unit S/m. In terms of microscopic quantities, ρ = m/(ne²τ).

Effect of stretching: if a wire is stretched to n times its length, its volume stays the same, so area becomes A/n and resistance becomes n² times.

Temperature dependence: for metals, ρ_T = ρ₀[1 + α(T – T₀)], where α is the temperature coefficient of resistivity; α is positive, so resistance rises with temperature (τ decreases due to more collisions). For semiconductors, resistivity falls with temperature because n increases. Alloys such as nichrome and manganin have very small α, so they are used in standard resistors.

Worked example: a copper wire (ρ = 1.7 × 10⁻⁸ Ω m) of length 10 m and area 1 mm² has R = 1.7 × 10⁻⁸ × 10 / 10⁻⁶ = 0.17 Ω.

Key points

• V = IR; R = ρL/A; σ = 1/ρ.
• Resistivity depends on material and temperature, not on shape.
• Stretching to n times length gives n² times resistance.
• Metals: α > 0. Semiconductors: resistivity decreases with temperature.

Exam tips

• Ratio questions on stretching and doubling length or area are very common.
• In V-I graphs, slope of V vs I gives R; slope of I vs V gives 1/R.

Common mistakes

• Saying resistivity changes when the wire is cut or stretched; only resistance does.
• Forgetting that the area also changes when a wire is stretched.
• Assuming every device obeys Ohm’s law.