Potential dividers

AS Level Physics · D.C. circuits. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.

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The idea

A potential divider is two resistors in series across a supply. They carry the same current, so the supply p.d. is shared in the ratio of their resistances. The bigger resistance takes the bigger share. The output is the p.d. across one of them.

If one resistor is a thermistor or an LDR, the output changes with temperature or light. When the resistance of the sensor falls, its share falls and the share of the fixed resistor rises.

A potentiometer is a uniform wire used as a divider. A sliding contact is moved until a galvanometer reads zero. At this balance point the p.d. across that length of wire equals the p.d. being measured.

Rules to remember

output p.d. = supply p.d. × R₁ ÷ (R₁ + R₂), where the output is across R₁
V₁/V₂ = R₁/R₂ for two resistors in series
LDR: brighter light, lower resistance, smaller p.d. across the LDR, larger p.d. across the fixed resistor
Thermistor (NTC): hotter, lower resistance, smaller p.d. across the thermistor, larger p.d. across the fixed resistor
Potentiometer: p.d. along a uniform wire is proportional to its length, so E₁/E₂ = L₁/L₂ (balance lengths)
Null method: at balance there is no current in the test cell, so no lost volts, and the p.d. measured equals its e.m.f.
A voltmeter of similar resistance connected across one resistor lowers the resistance of that part, so it reads less than expected

Common mistake

Students put the wrong resistance on top of the fraction. The resistance on top must be the one the output is taken across.

Worked example

A thermistor and a 2.0 kΩ fixed resistor are in series across a 9.0 V supply. The output is taken across the fixed resistor. What is the output when the thermistor has resistance 4.0 kΩ, and when it warms up and has resistance 1.0 kΩ?

Cold: V = 9.0 × 2.0 ÷ (2.0 + 4.0) = 9.0 × 2.0 ÷ 6.0 = 3.0 V.
Warm: V = 9.0 × 2.0 ÷ (2.0 + 1.0) = 9.0 × 2.0 ÷ 3.0 = 6.0 V.
The thermistor's resistance fell, so the fixed resistor took a larger share of the 9.0 V.

Answer: The output rises from 3.0 V to 6.0 V as the temperature rises.

Practice questions

Try each one, then open the answer.

1. A thermistor and a fixed resistor form a potential divider to operate a heater switch that must turn on when it gets cold. The output is taken across the thermistor. As temperature falls, the output p.d.

A
rises, because the thermistor's resistance rises and takes a larger share of the supply
B
falls
C
stays constant
D
becomes zero

Show answer

Answer: A. An NTC thermistor's resistance increases in the cold, so the p.d. across it increases and can trigger the switch.

2. A potential divider has resistors of 2.0 kΩ and 6.0 kΩ in series across 12 V. What is the p.d. across the 2.0 kΩ resistor?

A
1.5 V
B
3.0 V
C
4.0 V
D
9.0 V

Show answer

Answer: B. V = 12 × 2.0/(2.0 + 6.0) = 3.0 V.

3. An NTC thermistor and a fixed resistor are connected in series across a supply to make a fire alarm sensor. The output p.d. must rise as the temperature rises. Where should the output be taken?

A
across the thermistor
B
across the whole supply
C
across either, since both p.d.s rise
D
across the fixed resistor

Show answer

Answer: D. As temperature rises, the thermistor's resistance falls, so it takes a smaller share of the supply. The p.d. across the fixed resistor therefore rises.

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