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Theme B — The Particulate Nature of Matter

B.5.6 Electrical Resistivity — IB Physics HL Revision Notes & Topical Questions

5 min read 10 parts Lesson 47 of 53 1 quiz

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Lesson summary

B.5.6 Electrical Resistivity — IB PHYSICS SL notes: Use ρ = RA/L to relate the resistance of a wire to its dimensions and material.Three factors — the material, the length and the cross-sectional areaLength — resistance is directly proportional to the length of the conductorArea …

  • What the symbols mean
  • Constant of proportionality
  • Conductors
  • Insulators
  • Constantan
  • Power lines

Part 1 of 8What a sample's resistance depends on

Three factors
  • Three factors the material, the length and the cross-sectional area
Length
  • Length resistance is directly proportional to the length of the conductor
Area
  • Area resistance is inversely proportional to the cross-sectional area
  • The cross-sectional area of a wire is modelled as a circle, A = πr2 = πd2/4.
  • If the diameter doubles, the area quadruples and the resistance falls to one quarter.

Part 2 of 8Resistivity

1Resistivity
  • Resistivity a property of a material describing how strongly it opposes current
  • It equals the resistance per unit length of a sample of unit cross-sectional area.
  • It depends only on the material and its temperature, not on the shape of the sample.
2Equation
  • R = ρL / A,
    • or ρ = RA / L (data booklet)
      What the symbols mean: R = resistance (Ω) · ρ = resistivity (Ω m) · L = length (m) · A = cross-sectional area (m2)
  • R is resistance in Ω, L is length in m and A is area in m2.
  • ρ is resistivity in Ω m.
3Unit
  • Ω m is the unit of resistivity, not of resistance
4Constant of proportionality
  • ρ links the proportionalities R ∝ L and R ∝ 1/A

Part 3 of 8Resistivity of materials

Conductors
  • Conductors metals have low resistivity, so copper and aluminium make ideal wires
  • Copper has ρ = 1.7 × 10−8 Ω m.
  • Aluminium has ρ = 2.6 × 10−8 Ω m.
Insulators
  • Insulators plastic and rubber have such high resistivity that virtually no current flows
  • They house wires and prevent dangerous electric shocks.
Constantan
  • Constantan an alloy of resistivity about 5 × 10−7 Ω m that barely changes with temperature
  • It is used in standard resistors.
Nichrome
  • Nichrome an alloy of high resistivity used in heating elements
  • Much energy is transferred as heat.
Power lines
  • Power lines aluminium is used because it has low density as well as low resistivity

Part 4 of 8Wire area and stretching

1Circular wire
  • A = πr2, and area depends on the square of the diameter
  • A small error in d produces a large error in R.
  • For d = 0.50 mm, d = 5.0 × 10−4 m and A = 1.96 × 10−7 m2.
2Unit conversion
  • 1 mm2 is 10−6 m2, not 10−3 m2
3Stretching at constant volume
  • A decreases as L increases
  • The resistance increases by the factor (L2/L1)2.
4Comparing wires
  • Comparing wires write R2/R1 = (ρ2/ρ1)(L2/L1)(A1/A2) and substitute ratios

Part 5 of 8Practical to find resistivity

1
Fix the test wire along a metre rule.
2
Measure its diameter with a micrometer at several places and in two perpendicular directions, then average.
3
Connect the wire in series with a cell, ammeter and variable resistor, with a voltmeter across the wire.
4
For different lengths L between the contact points, record V and I and find R = V/I.
5
Plot R against L; the line is straight through the origin with gradient ρ/A.
6
Calculate ρ = gradient × A.
Heating
  • Heating use small currents and close the switch only briefly, because a temperature rise increases R
Zero error
  • Zero error check the micrometer for zero error before measuring

Part 6 of 8Uncertainties in the practical

Diameter
  • Diameter it is usually the largest source of uncertainty
  • The fractional uncertainty in A is twice that in d.
  • d = 0.50 ± 0.01 mm is 2%, so A has 4%.
Combining
  • Combining the fractional uncertainties in R, A and L add to give that in ρ
Plausibility
  • Plausibility a copper value far from 10−8 Ω m suggests a unit error in the diameter or area

Part 7 of 8Temperature and applications

Metals
  • Metals resistivity rises with temperature
Semiconductors
  • Semiconductors resistivity falls with temperature
Sensors
  • Sensors strain gauges, resistance thermometers and variable resistors use resistance changing with dimensions or temperature

Part 8 of 8More from the specification

More from the specification
Resistivity and Conductor Resistance
  • Electrical resistivity (ρ = RA/L) is an intrinsic material property independent of dimensions, relating resistance R to length and cross-sectional area.
Internal Resistance and Terminal PD
  • A real battery or cell has internal resistance r;
    • the terminal potential difference across connected components is given by V = ℰ - Ir.
Resistors in Parallel Combinations
  • When circuit components are connected in parallel, the total potential difference is identical across each branch while electric charge splits proportionally.
Temperature Characteristics of a Thermistor
  • For an NTC thermistor,
  • increasing temperature releases more mobile charge carriers, decreasing resistance as shown on a non-linear I-V graph.
Potential difference
  • Work done by a unit charge passing through a component.
Internal resistance
  • Resistance inside a cell that causes lost p.d.
Circuit diagram
  • Drawing of a circuit using standard symbols.
Light intensity
  • How bright the light falling on a surface is.
Fixed resistor
  • Resistor with a resistance that cannot change.
Conventional current
  • Current drawn from the positive to the negative terminal.
Charge
  • Quantity of electricity in coulombs, Q = It.
    What the symbols mean: Q = charge (C) · I = current (A) · t = time (s)
Parallel
  • Components on separate branches, each with the full e.m.f.
Component
  • Part of a vector along one axis.
Battery
  • Two or more cells joined together.
Thermistor
  • Resistor whose resistance falls as temperature rises.
Components
  • Perpendicular parts of a vector that add to give the vector.
Terminal
  • Connection point of a cell, labelled + or −.

Key terms

TermMeaning
Resistivityρ, material property in Ω m, defined by R = ρL/A.
Cross-sectional areaArea of the slice perpendicular to the current; πr^{2} for a round wire.
MicrometerInstrument measuring wire diameter to 0.01 mm.
ConstantanAlloy with a high resistivity that barely changes with temperature.
NichromeAlloy used in heating elements.

Avoid these mistakes

1
Using the diameter instead of the area, or forgetting to square the radius.
2
Writing resistivity in Ω instead of Ω m.
3
Not converting mm or mm^{2} to metres — 1 mm^{2} is 10^{−6} m^{2}.
4
Saying resistivity depends on the length — it is a material property.
5
Forgetting that the area also changes when a wire is stretched.

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