Short on time? This page is the recap. For the derivations, Figures 1 and 2 and the worked examples, use the full study guide.
Recap
- Resistance measures how strongly an object opposes the movement of charge. R = ΔV/I, in ohms (1 Ω = 1 V/A).
- Resistivity ρ (Ω·m) is a property of the material, set by its atomic and molecular structure. It does not depend on the shape of the sample.
- In a metal, resistivity usually rises with temperature, because carriers scatter more often off the more strongly vibrating ions.
- An ohmic element has constant resistance for all currents; in the ohmic model its resistivity does not change with temperature.
- Resistors convert electrical energy to thermal energy, which can warm the resistor and its surroundings.
Key relationships
| Situation | Relationship | Note |
|---|---|---|
| Any element, at one moment | R = ΔV/I | Definition |
| Uniform wire | R = ρℓ/A | From E = ρJ, J = I/A and ΔV = Eℓ |
| Round wire | A = πr² = πd²/4 | Halve the diameter first |
| Wire stretched to n times its length | R → n²R | Volume fixed, so A → A/n |
| Resistivity varies along the wire | R = (1/A)∫₀ˡ ρ(x) dx | Thin slices in series |
| Ohmic element | ΔV = IR with R constant | Ohm’s law |
| Graph of I (vertical) against ΔV | slope = 1/R | So R = 1/slope |
Assumptions behind the results
- The wire has a uniform cross-section, and the current is spread evenly across it.
- The current is steady.
- Unless told otherwise, resistors are ohmic, and connecting wires have negligible resistance.
Mistakes to avoid
- Calling R = ΔV/I “Ohm’s law”. Ohm’s law is the claim that R is constant.
- Diameter used as radius. This makes A four times too large.
- Stretched wire: R × n. It is R × n², because the area shrinks too.
- Slope of I against ΔV taken as R. It is 1/R.
- Tangent slope for a lamp. Use ΔV/I at the point.
- Single ρ when ρ varies. Integrate.
- Line through the first and last points. Draw a best-fit line through all the data.
Quick self-check
- A wire is 1.0 m long, with area 1.0 × 10⁻⁶ m² and ρ = 5.0 × 10⁻⁷ Ω·m. What is R? (0.50 Ω)
- An ohmic element’s I–ΔV graph has slope 0.10 A/V. What is R? (10 Ω)
- A uniform wire of resistance R is cut into two equal halves. What is the resistance of each half? (R/2)
Next: practice questions.