What you'll learn
This topic examines how conductors oppose the flow of electric current and how resistors control current in circuits. You'll master calculations using Ohm's law, analyse series and parallel arrangements, and interpret current-voltage characteristics. These concepts appear frequently in Paper 2 (Core) and Paper 4 (Extended), accounting for significant marks in both multiple-choice and structured questions.
Key terms and definitions
Resistance — the opposition to current flow in a conductor, measured in ohms (Ω). A higher resistance means less current flows for a given potential difference.
Resistor — a component specifically designed to provide a fixed or variable resistance in a circuit, controlling the current that flows.
Ohm's law — the principle stating that current through a conductor is directly proportional to the potential difference across it, provided temperature remains constant. Expressed as V = IR.
Ohmic conductor — a conductor that obeys Ohm's law, displaying constant resistance over a range of voltages. Its current-voltage graph is a straight line through the origin.
Series circuit — an arrangement where components are connected end-to-end in a single loop, so the same current flows through each component.
Parallel circuit — an arrangement where components are connected across common points, providing multiple paths for current. The potential difference across each parallel branch is identical.
Variable resistor (rheostat) — a resistor whose resistance can be adjusted, allowing control over current in a circuit.
Thermistor — a temperature-dependent resistor whose resistance decreases as temperature increases (for negative temperature coefficient types tested at IGCSE).
Core concepts
Understanding electrical resistance
Resistance occurs because electrons collide with ions in the conductor's lattice structure as they flow. Each collision transfers energy, converting electrical energy to thermal energy and impeding electron movement. The relationship between potential difference (V), current (I), and resistance (R) is given by:
V = IR
Where:
- V is potential difference in volts (V)
- I is current in amperes (A)
- R is resistance in ohms (Ω)
Rearranging gives R = V/I or I = V/R, depending on what you need to calculate.
Factors affecting resistance in a wire:
- Length — longer wires have greater resistance because electrons undergo more collisions
- Cross-sectional area — thinner wires have higher resistance because electrons are more crowded
- Material — different materials have different numbers of free electrons and lattice structures
- Temperature — higher temperatures increase resistance in metals as ions vibrate more vigorously
Ohmic and non-ohmic conductors
An ohmic conductor maintains constant resistance regardless of the potential difference applied. Metallic conductors at constant temperature behave as ohmic conductors. Their I-V characteristic is a straight line passing through the origin with gradient equal to 1/R.
Non-ohmic conductors do not obey Ohm's law. Their resistance changes with current or voltage:
Filament lamp — as current increases, the filament temperature rises dramatically. Higher temperature means more vigorous ion vibrations, increasing resistance. The I-V graph curves, showing reduced gradient (increased resistance) at higher voltages.
Diode — conducts readily in one direction (forward bias) but has extremely high resistance in the reverse direction. The I-V graph shows steep current increase above a threshold voltage (~0.7V for silicon) in forward bias, and near-zero current in reverse bias.
Thermistor (NTC type) — resistance decreases as temperature increases because more charge carriers become available for conduction. Used in temperature sensors and control circuits.
Light-dependent resistor (LDR) — resistance decreases as light intensity increases. Photons provide energy to release more charge carriers. Used in automatic lighting systems and light meters.
Resistors in series circuits
Components connected in series have:
- Identical current through each component (I₁ = I₂ = I₃...)
- Total potential difference shared between components (V_total = V₁ + V₂ + V₃...)
- Total resistance equals the sum of individual resistances:
R_total = R₁ + R₂ + R₃ + ...
Each resistor causes an additional voltage drop. Adding more resistors in series always increases total resistance, reducing circuit current. The largest resistor has the largest voltage across it, proportional to its resistance value.
Example: Three resistors of 2Ω, 3Ω, and 5Ω in series with a 12V battery:
- R_total = 2 + 3 + 5 = 10Ω
- Current = V/R = 12/10 = 1.2A (same everywhere)
- V₁ = IR₁ = 1.2 × 2 = 2.4V
- V₂ = IR₂ = 1.2 × 3 = 3.6V
- V₃ = IR₃ = 1.2 × 5 = 6.0V
- Check: 2.4 + 3.6 + 6.0 = 12V ✓
Resistors in parallel circuits
Components connected in parallel have:
- Same potential difference across each component (V₁ = V₂ = V₃ = V_total)
- Total current split between branches (I_total = I₁ + I₂ + I₃...)
- Combined resistance less than the smallest individual resistance:
1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + ...
For two resistors only, this simplifies to:
R_total = (R₁ × R₂)/(R₁ + R₂)
Adding resistors in parallel always decreases total resistance because you create additional pathways for current. Each branch operates independently — current in one branch doesn't affect others.
Example: Two resistors of 6Ω and 3Ω in parallel with a 12V supply:
- 1/R_total = 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2
- R_total = 2Ω
- I₁ = V/R₁ = 12/6 = 2A
- I₂ = V/R₂ = 12/3 = 4A
- I_total = 2 + 4 = 6A
- Check using Ohm's law: I_total = V/R_total = 12/2 = 6A ✓
Variable resistors and their applications
A variable resistor has an adjustable slider that changes the length of wire in the circuit. Moving the slider alters resistance, thereby controlling current. Applications include:
- Volume controls in audio equipment
- Brightness controls for lamps
- Temperature adjustment in heating systems
Potential divider circuits use two resistors in series to produce a desired output voltage that is a fraction of the input voltage:
V_out = (R₂/(R₁ + R₂)) × V_in
Where V_out is measured across R₂. Replacing one fixed resistor with a thermistor or LDR creates temperature or light sensors that produce voltage outputs varying with environmental conditions. These voltage changes can trigger other circuit components.
Practical measurements of resistance
To measure resistance experimentally:
- Connect the resistor in series with an ammeter and power supply
- Connect a voltmeter in parallel across the resistor
- Record current (I) and potential difference (V)
- Calculate R = V/I
- Repeat with different supply voltages to verify consistency (for ohmic conductors)
- Plot I-V graphs to identify conductor type
For testing Ohm's law, adjust supply voltage systematically and plot current against voltage. An ohmic conductor produces a straight line through the origin. The gradient equals 1/R.
Worked examples
Example 1: Series circuit calculation
Question: A 9V battery is connected to three resistors in series: 10Ω, 15Ω, and 20Ω.
(a) Calculate the total resistance of the circuit. [1] (b) Calculate the current flowing through the circuit. [2] (c) Calculate the potential difference across the 15Ω resistor. [2]
Solution:
(a) R_total = R₁ + R₂ + R₃ R_total = 10 + 15 + 20 = 45Ω [1]
(b) Using V = IR, rearrange to I = V/R I = 9/45 = 0.2A [1] with correct working [1]
(c) V = IR [1] V = 0.2 × 15 = 3V [1]
Example 2: Parallel circuit calculation
Question: Two resistors are connected in parallel to a 12V supply. One resistor has resistance 4Ω, the other 6Ω.
(a) Calculate the combined resistance. [2] (b) Calculate the total current drawn from the supply. [2] (c) Explain why the total resistance is less than either individual resistor. [2]
Solution:
(a) 1/R_total = 1/R₁ + 1/R₂ [1] 1/R_total = 1/4 + 1/6 = 3/12 + 2/12 = 5/12 R_total = 12/5 = 2.4Ω [1]
(b) I = V/R [1] I = 12/2.4 = 5A [1]
(c) Each resistor provides a separate path for current [1] Adding parallel branches increases total current flow for the same voltage, which means overall resistance must decrease [1]
Example 3: Variable resistance application
Question: A potential divider consists of a fixed 200Ω resistor and a thermistor in series with a 6V supply. At room temperature, the thermistor has resistance 100Ω.
(a) Calculate the output voltage across the thermistor at room temperature. [3] (b) Describe and explain what happens to the output voltage when temperature increases. [3]
Solution:
(a) V_out = (R_thermistor/(R_fixed + R_thermistor)) × V_in [1] V_out = (100/(200 + 100)) × 6 [1] V_out = (100/300) × 6 = 2V [1]
(b) The thermistor resistance decreases as temperature increases [1] This reduces the proportion of total resistance that the thermistor represents [1] Therefore the output voltage across the thermistor decreases [1]
Common mistakes and how to avoid them
• Mistake: Confusing series and parallel resistance formulae — using addition for parallel circuits or reciprocals for series circuits. Correction: Remember "series is simple addition" (R_total = R₁ + R₂). For parallel, use reciprocals (1/R_total = 1/R₁ + 1/R₂) or the product-over-sum formula for two resistors only.
• Mistake: Assuming voltage splits equally in series circuits regardless of resistance values. Correction: Voltage divides proportionally to resistance. A 10Ω resistor receives twice the voltage of a 5Ω resistor in the same series circuit because V = IR and I is constant.
• Mistake: Stating that resistance is directly proportional to voltage or current without mentioning constant temperature. Correction: Ohm's law requires constant temperature. For conductors like filament lamps, resistance increases with current due to temperature rise. Always specify that temperature must remain constant for ohmic behaviour.
• Mistake: Believing that adding a resistor in parallel increases circuit resistance. Correction: Parallel resistors always decrease total resistance by providing additional current paths. The combined resistance is always less than the smallest individual resistor in the parallel arrangement.
• Mistake: Forgetting to convert units — particularly using mA as if it were A, or kΩ as if it were Ω. Correction: Convert all currents to amperes (divide mA by 1000) and all resistances to ohms (multiply kΩ by 1000) before substituting into formulae. Show conversions explicitly in working.
• Mistake: Drawing I-V graphs with current on the x-axis instead of voltage. Correction: Standard I-V characteristic graphs have voltage (the independent variable) on the x-axis and current (the dependent variable) on the y-axis. The gradient represents 1/R, not R.
Exam technique for "Resistance and resistors"
• Command word "Calculate" requires numerical working and a final answer with correct units. Show the formula, substitution with units, and final answer. Marks are typically split: 1 mark for method/formula, 1 mark for correct answer. For 3-mark calculations, expect intermediate steps to be marked separately.
• When drawing or interpreting I-V graphs, remember that ohmic conductors give straight lines through the origin. Clearly label axes with quantities and units. For filament lamps, the curve should show decreasing gradient (increasing resistance) at higher voltages. Diodes show a threshold voltage before conducting.
• "Explain" questions about resistance changes require references to both charge carrier behaviour and energy transfers. State what happens to resistance, link this to the I = V/R relationship, then describe the physical mechanism (collision frequency, carrier availability, temperature effects).
• Circuit calculations often appear in 6-8 mark structured questions building through parts (a) to (d). Early parts typically test recall (definitions, 1-2 marks), middle parts require calculations (3-4 marks total), and final parts demand explanations or predictions (2-3 marks). Budget roughly 1 minute per mark.
Quick revision summary
Resistance opposes current flow, measured in ohms using R = V/I. Ohmic conductors maintain constant resistance; non-ohmic types (filament lamps, diodes, thermistors) show varying resistance. In series circuits: same current everywhere, voltages add, R_total = R₁ + R₂. In parallel circuits: same voltage across branches, currents add, 1/R_total = 1/R₁ + 1/R₂. Series arrangements increase total resistance; parallel arrangements decrease it. Potential dividers split voltage proportionally. Thermistors decrease resistance with temperature; LDRs decrease resistance with light intensity. Master formula rearrangements and unit conversions for calculation questions.