Resistance, Ohm's law & I–V graphs

Learning goals

What students should be able to explain, calculate and investigate after completing Resistance, Ohm's Law, Resistivity and I-V Graphs.

• Define resistance and state Ohm's law.
• Define resistivity and calculate the resistance of a wire from its dimensions.
• Interpret I-V graphs for an ohmic conductor, a filament lamp, and a diode.
• Describe an experiment to verify Ohm's law and to investigate I-V characteristics.

Key definitions

Precise terms, symbols and SI units used in Resistance, Ohm's Law, Resistivity and I-V Graphs.

Resistance and Ohm's Law

Resistance measures how much a component opposes current for a given potential difference: R = V/I.

Ohm's law is a special case where this ratio stays constant regardless of the applied voltage or current — this holds for many resistors at constant temperature, but not for all components (e.g. a filament lamp or a diode, where resistance changes with conditions).

Resistivity

Resistivity is a property of a material at a stated temperature. A wire’s resistance also depends on its length and cross-sectional area.

For a uniform wire, R = ρl/A. A longer wire has greater resistance and a wire with greater cross-sectional area has lower resistance. Resistivity depends on the material and on physical conditions, especially temperature; it is not an unchanging value for copper under all conditions.

I-V Characteristics

An ohmic conductor (e.g. a resistor at constant temperature) gives a straight-line I-V graph through the origin — its resistance (found from V/I, or from 1/slope) is constant.

In a metal, an electric field produces a slow drift of electrons. As they move, electrons scatter from the vibrating lattice and transfer energy to it. When a filament gets hotter, lattice vibrations increase, collisions become more frequent and its resistance usually rises. This is why a filament-lamp I–V graph curves: current no longer rises in direct proportion to voltage.

• Electric field produces electron drift.
• Electrons scatter from the lattice and transfer energy to it.
• Higher temperature means stronger lattice vibrations.
• A hot metal filament therefore usually has greater resistance.

Diagrams and formulae

Key diagrams and equations for reviewing Resistance, Ohm's Law, Resistivity and I-V Graphs.

• I-V graphs: an ohmic conductor gives a straight line through the origin (constant resistance); a filament lamp curves as its resistance increases with temperature; a diode conducts strongly only above a threshold voltage in one direction.
• Resistivity: resistance depends on the material (ρ), the length (l), and the cross-sectional area (A) of the sample.

I-V graphs: an ohmic conductor gives a straight line through the origin (constant resistance); a filament lamp curves as its resistance increases with temperature; a diode conducts strongly only above a threshold voltage in one direction.

Resistivity: resistance depends on the material (ρ), the length (l), and the cross-sectional area (A) of the sample.

• Key relationships
• Ohm's Law: V = IR (R constant, for an ohmic conductor at constant temperature)

Worked Example

A step-by-step calculation applying the central relationships in Resistance, Ohm's Law, Resistivity and I-V Graphs.

A wire of resistivity 1.7 × 10⁻⁸ Ω m has a length of 2 m and a cross-sectional area of 4 × 10⁻⁷ m². Calculate its resistance.

• R = ρl/A
• R = (1.7 × 10⁻⁸ × 2) ÷ (4 × 10⁻⁷)
• R = (3.4 × 10⁻⁸) ÷ (4 × 10⁻⁷) = 0.085 Ω

Practical / Investigation

Experimental planning, measurement and evaluation for Resistance, Ohm's Law, Resistivity and I-V Graphs.

Experiment: Verifying Ohm's Law / investigating I-V characteristics

• Apparatus: power supply (variable), component under test (resistor, filament lamp, or diode), ammeter, voltmeter, rheostat (variable resistor) to vary current.
• Independent variable: potential difference across the component (varied using the rheostat or supply). Dependent variable: current through the component.
• Method: for each of several different settings, record the current and the potential difference across the component; for a diode, repeat with the connections reversed to investigate reverse bias.
• Plot I (y-axis) against V (x-axis). A straight line through the origin confirms Ohm's law (ohmic behaviour); a curve indicates non-ohmic behaviour.
• Sources of error: the component may heat up during the experiment, changing its resistance during data collection (especially for a filament lamp); meter resistance affects the circuit slightly.
• Improvement: take readings quickly, or allow the component to cool between readings, to reduce self-heating effects.

• Apparatus: uniform resistance wire, metre stick, micrometer, ohmmeter/multimeter and crocodile clips.
• Method: measure the resistance R of a known length l. Measure the wire diameter at several positions and in two perpendicular directions; find the mean diameter and calculate A = πd²/4.
• Calculation: ρ = RA/l, with R in Ω, A in m² and l in m, giving resistivity in Ω m.
• Precautions: check meter/lead zero error, avoid warming the wire, keep contact points clean, and use a long length so percentage uncertainty in l is small.

• For a metallic conductor, place the coil and thermometer in a stirred water bath. Measure resistance at several temperatures while allowing the wire and thermometer to reach thermal equilibrium; plot R against temperature.
• For an NTC thermistor, repeat with the thermistor in good thermal contact with the water. Its resistance decreases as temperature increases.
• Use a low measuring current to limit self-heating, stir the bath, and take readings only when the temperature is steady.

• Connect the ammeter in series and voltmeter across the component; include a rheostat/current limiter and begin at a low supply voltage.
• Change potential difference through a safe range, record paired V and I readings, and plot I against V with both units shown.
• For an ohmic resistor keep temperature approximately constant. Reverse polarity for negative-voltage data only where the component and meters permit.

Quick-Check Questions

Short retrieval questions covering the essential ideas in Resistance, Ohm's Law, Resistivity and I-V Graphs.

1. State Ohm's law.

2. Define resistivity and give its unit.

3. Why does a filament lamp's I-V graph curve, rather than being a straight line?

5. In which direction does a diode conduct easily?

Common mistakes and exam tips

Common misconceptions and exam-technique checks for Resistance, Ohm's Law, Resistivity and I-V Graphs.

• Avoid: Assuming every component obeys Ohm’s law. Correct idea: Ohm’s law applies when V is proportional to I under constant physical conditions; filament lamps and diodes are non-ohmic.
• Avoid: Confusing resistance with resistivity. Correct idea: Resistance describes a particular component in Ω; resistivity is a material property in Ω m at a stated temperature.
• Avoid: Using the gradient of I against V as the resistance. Correct idea: For an I–V graph, gradient is conductance for an ohmic component; at any point R = V/I.

• Convert wire length to metres and cross-sectional area to m² before using R = ρl/A.
• State whether a diode is forward or reverse biased.

Exam-style questions

Five distinct exam-style questions with worked answers on Resistance, Ohm's Law, Resistivity and I-V Graphs.

Higher Level — Required Content

Required Higher Level content for this topic.

Use V = IR with the circuit rules: current is common in series; potential difference is common in parallel.

End-of-unit revision

Formulae, definitions, experiments and mixed revision questions for Resistance, Ohm's Law, Resistivity and I-V Graphs.

Try without notes: sketch I-V graphs for an ohmic conductor, a filament lamp, and a diode, and state one feature of each.

• R = V/I
• V = IR (Ohm's Law, constant R)
• R = ρl/A

• Resistance
• Ohm's Law
• Ohmic conductor
• Resistivity

• I-V graphs for an ohmic conductor, a filament lamp, and a diode
• Resistivity diagram showing length and cross-sectional area

• Verifying Ohm's law / investigating the I-V characteristic of a component

• Assuming all components are ohmic
• Confusing resistivity with resistance
• Reading resistance from the slope of a curved I-V graph

1. A resistor has a potential difference of 6 V across it when a current of 0.4 A flows. Calculate its resistance.

2. A copper wire has resistance 0.2 Ω. A second copper wire, twice as long but with half the cross-sectional area, is made from the same copper. Calculate its resistance.

3. Explain why a diode is a useful component for converting alternating current to direct current (rectification).

1. R = V/I = 6 ÷ 0.4 = 15 Ω.

2. R = ρl/A; doubling l doubles R, halving A doubles R again, so the new resistance = 0.2 × 2 × 2 = 0.8 Ω.

3. A diode only allows current to flow in one direction, so when placed in an AC circuit it blocks current during the 'reverse' half of each cycle, allowing only one-directional (direct) current to pass through.

4. With equal resistors, = 9 × R/(2R) = 4.5 V.

The table below shows which equations from this unit appear in the official State Examinations Commission Formulae and Tables book, and where.