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Wire resistance and temperature

Key idea: A wire's resistance is proportional to its length and inversely proportional to its cross-sectional area. Doubling the diameter quadruples the area, so the resistance falls to a quarter.

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Two wires made of the same metal can have very different resistances. A long thin wire resists the current more than a short thick one, and either wire resists more when it is hot. This lesson compares wires by their length, their thickness and their temperature, one at a time.

Length and cross-sectional area

For a uniform wire of the same metal at the same temperature:

What changesWhat stays the sameEffect on resistance
length Lcross-sectional area AR ∝ L
cross-sectional area Alength LR ∝ 1/A
Separate length and cross-sectional area comparisonsTwo panels compare uniform wires of the same material at the same temperature. The left shows two lengths, L and 2L, with equal cross-sectional area: their resistances are R and 2R. The right shows enlarged circular end views with diameters d and 2d at equal wire length: areas are A and 4A, so resistances are R and R divided by four. The end views share a scale within their panel; they do not show wire length.Same material and temperature in both comparisonsChange lengthKeep cross-sectional area fixed.L2LR2RTwice the length → twice the resistanceChange diameterKeep wire length fixed.Enlarged circular end viewsd2darea Aresistance Rarea 4Aresistance R/4Twice the diameter → four times the area
Separate length and cross-sectional area comparisonsTwo panels compare uniform wires of the same material at the same temperature. The left shows two lengths, L and 2L, with equal cross-sectional area: their resistances are R and 2R. The right shows enlarged circular end views with diameters d and 2d at equal wire length: areas are A and 4A, so resistances are R and R divided by four. The end views share a scale within their panel; they do not show wire length.Same material and temperature in both comparisonsChange lengthKeep cross-sectional area fixed.L2LR2RTwice the length → twice the resistanceChange diameterKeep wire length fixed.Enlarged circular end viewsd2darea Aresistance Rarea 4Aresistance R/4Twice the diameter → four times the area
Compare one variable at a time, using the same material and temperature. Double length at fixed area: resistance doubles. Double diameter at fixed length: area becomes four times as large and resistance becomes one quarter as large.

Why? In a longer wire each electron drifts through more of the lattice and collides with more ions, so twice the length gives twice the resistance. A thicker wire is like several thin wires side by side: more paths for the charge, so more current for the same p.d. and a smaller resistance. Twice the area gives half the resistance.

Each rule holds only while the other quantity stays fixed. If both length and area change, apply each factor in turn.

Exam-style question 1

Compare two wires of different length

3 marks

Examination question

Wires A and B are made of the same metal, are at the same temperature and have the same cross-sectional area. A is twice as long as B. Compare their resistances and explain the difference in terms of electrons. [3 marks]

State the relationship, the ratio and the reason

Work on paper or in the box; compare the original answer directly with the model and criteria.

Show solution step by step

  1. Apply the proportionality

    2 marks

    Method

    Resistance is proportional to length, so A has twice the resistance of B.

    Reason

    Material, temperature and area are the same.

    Working

    Answer

    RA/RB = LA/LB = 2

  2. Explain the difference

    1 mark

    Method

    Electrons in A drift through more of the lattice and make more collisions with ions.

    Reason

    More collisions oppose the flow of charge more.

    Working

    Answer

    Longer path → more collisions → larger R

Diameter is not area

Questions often give a wire’s diameter d. Its cross-section is a circle of area π d²/4, so the area is proportional to the square of the diameter: A ∝ d². Doubling the diameter gives four times the area, and so one quarter of the resistance, not one half.

Worked example 1

Change the length and the diameter

Problem

A wire has a resistance of 2.0 Ω. A second wire of the same metal, at the same temperature, is three times as long and has twice the diameter. Find its resistance.

Show full solution

  1. Apply the length factor

    Method

    Three times the length gives three times the resistance.

    Reason

    R ∝ L when the area is fixed.

    Working

    Answer

    × 3

  2. Apply the area factor

    Method

    Twice the diameter gives four times the area, so one quarter of the resistance.

    Reason

    A ∝ d² and R ∝ 1/A.

    Working

    Answer

    × 1/4

  3. Combine the factors

    Working

    Answer

    R = 2.0 Ω × 3 × 1/4 = 1.5 Ω

Check your understanding 1 Check the area before the resistance

A wire has resistance 6.0 Ω. Another wire of the same metal and temperature is twice as long and has half the diameter. Find its cross-sectional area as a fraction of the first wire’s area, then its resistance.

Show the steps Optional working
Unit: Ω
Show answer

The area ratio is (1/2)² = 1/4 = 0.25. The new resistance is 6.0 × 2/(1/4) = 48 Ω: twice the length and one quarter of the area each increase the resistance.

Mind stretcher 1 Stretch a wireExtension

A wire is stretched evenly to twice its original length. Its volume, material and temperature do not change. Enter the new resistance divided by the original resistance. Use the unchanged volume to work out what happens to the area first.

Show the steps Optional working
Show answer

The volume is AL. If the volume stays the same and L doubles, A must halve. Both changes increase the resistance:

R_new = R × 2 × 2 = 4R

The resistance becomes four times the original.

A hotter metal has a larger resistance

When a metal gets hotter, its ions vibrate more. The drifting electrons collide with them more often, so the resistance rises. At the same p.d., the hot wire carries less current, because I = V/R.

This is why a filament lamp’s I–V graph curves as the filament heats up. It is also why a fair comparison of two wires keeps them at the same temperature: in an experiment, use a small current and switch off between readings so that the wire does not warm up. The electrical investigations practical shows how to measure resistance safely.

Check your understanding 2 Change what the supply keeps fixed

A metal wire warms up, but the supply is adjusted to keep its current unchanged. Must the p.d. across the wire rise, fall or stay the same? Explain using the temperature change and V = IR. Work on paper or in the box.

Show answer

The hotter metal has a larger resistance because ions vibrate more and electrons collide with them more often. At fixed current, V = IR requires a larger p.d. Compare both decisions: resistance rises, and this time current is fixed rather than p.d. The earlier conclusion of a smaller current applied only at fixed p.d.

Common mistakes

  • Saying “longer means more resistance” when the area has changed too. Apply each factor separately.
  • Halving the resistance for a doubled diameter; the area, and so the factor, is four.
  • Comparing wires at different temperatures.

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Syllabus and review details

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