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3.2.4 Ohm's law

The resistance equation

  1. Potential difference, current and resistance are linked by the equation V=IRV = IRV=IR, which you must be able to recall and apply.
    1. VVV is the potential difference in volts, V\text{V}V.
    2. III is the current in amperes, A\text{A}A.
    3. RRR is the resistance in ohms, Ω\OmegaΩ.
  2. The equation can be rearranged to find whichever quantity is unknown.
    1. To find the current, divide the potential difference by the resistance, I=VRI = \dfrac{V}{R}I=RV​.
    2. To find the resistance, divide the potential difference by the current, R=VIR = \dfrac{V}{I}R=IV​.
  3. Values given in milliamperes or kilohms must be changed to amperes and ohms before they are substituted.
  4. For a component whose resistance does not change, the equation shows that doubling the potential difference doubles the current.
  5. For a fixed potential difference, doubling the resistance halves the current.

Constant resistance

Definition

Ohm's law

The current through a conductor at constant temperature is directly proportional to the potential difference across it.

  1. For some resistors, the resistance stays the same as the current through them changes.
  2. A fixed resistor, or a metal wire kept at a constant temperature, behaves in this way.
  3. For these components, every pair of readings gives the same value of VI\dfrac{V}{I}IV​, so the current is directly proportional to the potential difference.
    1. This behaviour is described by Ohm's law, and a component that obeys it is called an ohmic conductor.
  4. The condition at constant temperature is part of the law, because heating a metal changes its resistance.

Changing resistance

  1. For other components, the resistance changes as the current changes.
  2. The resistance of a filament lamp increases as the current through it increases.
  3. The resistance of an NTC thermistor decreases as it gets warmer, and the resistance of an LDR decreases as more light falls on it.
  4. For these components, the value of VI\dfrac{V}{I}IV​ is different for different pairs of readings, so the current is not directly proportional to the potential difference.
  5. To decide whether a component has a constant resistance, calculate R=VIR = \dfrac{V}{I}R=IV​ for several pairs of readings and compare the answers.
    1. If the values agree, allowing for small measurement errors, the resistance is constant.
    2. If the values rise or fall steadily as the current increases, the resistance is changing.

Resistance of a wire

  1. The resistance of a metal wire depends on its length, its thickness, the material it is made from and its temperature.
  2. A longer wire has a larger resistance, because each electron makes more collisions with ions as it travels along it.
  3. For a wire of one material and thickness at constant temperature, the resistance is directly proportional to the length.
    1. Doubling the length of the wire doubles its resistance.
  4. A thinner wire has a larger resistance than a thicker wire of the same length and material.
Example

Finding a current

  • A resistor of 12 Ω12\ \Omega12 Ω is connected to a 3.0 V3.0\ \text{V}3.0 V cell.
I=VR I = \dfrac{V}{R} I=RV​ I=3.0 V12 Ω=0.25 A I = \dfrac{3.0\ \text{V}}{12\ \Omega} = 0.25\ \text{A} I=12 Ω3.0 V​=0.25 A
  • The same resistor is then connected to a 6.0 V6.0\ \text{V}6.0 V battery.
I=6.0 V12 Ω=0.50 A I = \dfrac{6.0\ \text{V}}{12\ \Omega} = 0.50\ \text{A} I=12 Ω6.0 V​=0.50 A
  • Doubling the potential difference has doubled the current, because the resistance of the resistor is constant.
Example

Finding a potential difference

  • A current of 2.5 mA2.5\ \text{mA}2.5 mA flows through a 4.8 kΩ4.8\ \text{k}\Omega4.8 kΩ resistor.
  • Both values are first changed to base units.
I=2.5÷1000=0.0025 A I = 2.5 \div 1000 = 0.0025\ \text{A} I=2.5÷1000=0.0025 A R=4.8×1000=4800 Ω R = 4.8 \times 1000 = 4800\ \Omega R=4.8×1000=4800 Ω V=IR=0.0025 A×4800 Ω=12 V V = IR = 0.0025\ \text{A} \times 4800\ \Omega = 12\ \text{V} V=IR=0.0025 A×4800 Ω=12 V
Practical

Resistance and the length of a wire

  • Aim: to investigate how the resistance of a wire depends on its length.
  • Apparatus: a 1.00 m1.00\ \text{m}1.00 m length of thin constantan wire taped flat along a metre rule, two crocodile clips, a single 1.5 V1.5\ \text{V}1.5 V cell or a power supply set to 2 V2\ \text{V}2 V, a switch, a digital ammeter, a digital voltmeter and connecting leads.
  • Method, setting up:
    • Clip one crocodile clip to the wire exactly at the 0 cm0\ \text{cm}0 cm mark of the ruler, and connect the cell, switch, ammeter and wire in series.
    • Fix the second crocodile clip at 20.0 cm20.0\ \text{cm}20.0 cm from the first, so that a 20.0 cm20.0\ \text{cm}20.0 cm length of wire is in the circuit.
    • Connect the voltmeter across the length of wire between the two clips.
  • Method, taking readings:
    • Close the switch, record the current and the potential difference, and open the switch again straight away.
    • Move the second clip to 40.0 cm40.0\ \text{cm}40.0 cm, 60.0 cm60.0\ \text{cm}60.0 cm, 80.0 cm80.0\ \text{cm}80.0 cm and 100.0 cm100.0\ \text{cm}100.0 cm, taking a pair of readings at each length.
    • Repeat the whole set of lengths twice more and calculate the mean resistance at each length using R=VIR = \dfrac{V}{I}R=IV​.
    • Plot a graph of resistance on the vertical axis against length on the horizontal axis and draw a line of best fit.
  • Variables: the independent variable is the length of wire, the dependent variable is its resistance, and the control variables are the material and thickness of the wire and its temperature.
  • Results: the graph is a straight line through the origin, with the resistance rising by roughly the same amount for every extra 20.0 cm20.0\ \text{cm}20.0 cm of wire.
  • Conclusion: the resistance of the wire is directly proportional to its length.
  • Watch out: if the first clip is not exactly at the zero mark, every length is wrong by the same amount and the line misses the origin, and short lengths draw a large current that warms the wire and raises its resistance.
  • Safety: the wire can become hot enough to burn, so keep the current small, open the switch between readings and do not touch the wire while the switch is closed.
Common Mistake
  • Do not forget to convert mA\text{mA}mA and kΩ\text{k}\OmegakΩ before substituting, because an unconverted value gives an answer that is wrong by a factor of 100010001000.
  • Do not assume every component obeys Ohm's law, because the resistance of a filament lamp, a thermistor and an LDR all change.
  • Do not describe Ohm's law without the condition of constant temperature.
Exam technique

Resistance calculations

  • Write the equation first, then show the substitution with units, then give the final answer with its unit, because each stage can earn a mark.
  • When you rearrange, rearrange the symbols before you put in any numbers, which makes slips easier to spot.
  • When a question gives a table of readings and asks whether the resistance is constant, calculate VI\dfrac{V}{I}IV​ for at least two rows and quote the values in your answer.
  • Give your answer to the same number of significant figures as the least precise value in the question, which is usually two or three.
Self review
  • Write down the equation linking potential difference, current and resistance.
  • Calculate the resistance of a lamp that carries 0.40 A0.40\ \text{A}0.40 A when it has 6.0 V6.0\ \text{V}6.0 V across it.
  • State Ohm's law.
  • Describe how to use a table of readings to show whether a component has a constant resistance.
  • Explain why a longer wire has a larger resistance.
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3.2.4 Ohm's law Revision Guide

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Revision notes for OCR GCSE Physics 3.2.4 Ohm's law: explanations and worked examples.

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