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
Curriculum Equation Key قوانين الرقم الهيدروجيني والاتزان الأيوني - pH & Acid-Base Equilibrium FormulasFast Exam Solution
pH = -log[H⁺] | pOH = -log[OH⁻] | pH + pOH = 14 | [H⁺][OH⁻] = 10⁻¹⁴
Word Equation & Conditions: الرقم الهيدروجيني = -لوغاريتم تركيز أيون الهيدروجين | pH + pOH = 14
Location & Specifications: الاتزان الأيوني، كيمياء 3 ثانوي، التحصيلي، و AP Chemistry.
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
Section 1 of 10What 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.
Section 2 of 10Resistivity
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)
- or ρ = RA / L (data booklet)
- 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
Section 3 of 10Resistivity 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
Section 4 of 10Wire 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
Section 5 of 10Practical 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
Section 6 of 10Uncertainties 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
Section 7 of 10Temperature 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
Section 8 of 10More 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 −.
Section 9 of 10Key terms
| Term | Meaning |
|---|---|
| Resistivity | ρ, material property in Ω m, defined by R = ρL/A. |
| Cross-sectional area | Area of the slice perpendicular to the current; πr^{2} for a round wire. |
| Micrometer | Instrument measuring wire diameter to 0.01 mm. |
| Constantan | Alloy with a high resistivity that barely changes with temperature. |
| Nichrome | Alloy used in heating elements. |
Section 10 of 10Exam technique
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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