Potential dividers
Section: D.C. Circuits | Syllabus: Cambridge AS Level Physics (9702)
The Potential Divider Principle
A potential divider is a circuit that uses two or more resistors in series to divide an input voltage into smaller output voltages. The output voltage depends on the ratio of the resistances.
Potential Divider A circuit arrangement of resistors in series that divides a supply voltage in proportion to the resistance values. FIG 10.11: Basic Potential Divider CircuitShow a circuit with a voltage source V_in connected to two resistors R₁ and R₂ in series. The output voltage V_out is taken across R₂ (between the junction of R₁ and R₂, and the negative terminal). Label V_in across the whole circuit and V_out across R₂. Show a high-resistance voltmeter connected across R₂ to measure V_out.
How It Works
Since R_1 and R_2 are in series:
- The same current I flows through both resistors
- The total resistance is R_T = R_1 + R_2
- The input voltage divides between the resistors in proportion to their resistances
Applying V = IR to each resistor:
V_1 = IR_1 and V_2 = IR_2The ratio of voltages equals the ratio of resistances:
(V_1/V_2) = (IR_1/IR_2) = (R_1/R_2)The Potential Divider Equation
The output voltage across R_2 can be expressed as a fraction of the input voltage:
Potential Divider Equation V_out = (R_2/R_1 + R_2) × V_inwhere V_out is measured across R_2
Derivation
The current through the circuit is:
I = V_inR_1 + R_2The voltage across R_2 is:
V_out = IR_2 = V_inR_1 + R_2 × R_2 = (R_2/R_1 + R_2) × V_in Key InsightThe output voltage is always a fraction of the input voltage. The larger the proportion of total resistance that R_2 represents, the larger the fraction of input voltage that appears across it.
Worked Example: Basic Potential DividerProblem: A 9 V battery is connected to a potential divider with R_1 = 700\,Ω and R_2 = 300\,Ω. Calculate the output voltage across R_2.
Solution:
V_out = (R_2/R_1 + R_2) × V_in V_out = (300/700 + 300) × 9 V_out = (300/1000) × 9 = 0.3 × 9 = 2.7 VAssumptions in Ideal Potential Dividers
- The voltmeter measuring V_out has infinite resistance (draws no current)
- No current is drawn from the output terminals by any connected circuit
- If current is drawn from the output, the output voltage will be less than calculated
Sensors in Potential Dividers
Potential dividers become particularly useful when one resistor is replaced by a sensor whose resistance varies with environmental conditions. This creates an output voltage that depends on temperature or light intensity.
Thermistors (NTC Type)
An NTC (Negative Temperature Coefficient) thermistor has a resistance that decreases as temperature increases.
NTC Thermistor A temperature-sensitive resistor whose resistance decreases non-linearly with increasing temperature. FIG 10.12: Thermistor in a Potential DividerShow two circuits side by side: Circuit A: Thermistor at the top (in R₁ position), fixed resistor at bottom. Label "V_out increases as temperature rises" because the thermistor resistance decreases, so more voltage appears across the fixed resistor. Circuit B: Fixed resistor at top, thermistor at bottom (in R₂ position). Label "V_out decreases as temperature rises" because the thermistor resistance (R₂) decreases, reducing V_out.
Worked Example: Thermistor Potential DividerProblem: A thermistor has resistance 2 kΩ at 20°C and 500 Ω at 50°C. It is connected in series with a 1 kΩ fixed resistor across a 9 V supply. The output is taken across the fixed resistor. Calculate the output voltage at (a) 20°C and (b) 50°C.
Solution:
(a) At 20°C: Thermistor resistance = 2 kΩ V_out = (1000/2000 + 1000) × 9 = (1000/3000) × 9 = 3 V (b) At 50°C: Thermistor resistance = 500 Ω V_out = (1000/500 + 1000) × 9 = (1000/1500) × 9 = 6 VThe output voltage increases from 3 V to 6 V as temperature rises, because the thermistor resistance decreases.
Light-Dependent Resistors (LDRs)
An LDR has a resistance that decreases as light intensity increases.
Light-Dependent Resistor (LDR) A light-sensitive resistor whose resistance decreases as the intensity of light falling on it increases. Resistance can range from ~100 Ω in bright light to ~1 MΩ in darkness. FIG 10.13: LDR in a Potential DividerShow two configurations: Configuration 1: LDR at top, fixed resistor at bottom - V_out increases in bright light (for circuits that activate in light). Configuration 2: Fixed resistor at top, LDR at bottom - V_out decreases in bright light (for circuits that activate in darkness). Include a note showing that this can control automatic lighting systems.
Worked Example: LDR Potential DividerProblem: An LDR has resistance 230 Ω in daylight and 1 MΩ in darkness. It is placed in series with a 10 kΩ resistor across a 9 V supply. The output is taken across the LDR. Calculate V_out in (a) daylight and (b) darkness.
Solution:
(a) In daylight: LDR resistance = 230 Ω V_out = (230/10000 + 230) × 9 = (230/10230) × 9 = 0.20 V (b) In darkness: LDR resistance = 1 MΩ = 1,000,000 Ω V_out = (1000000/10000 + 1000000) × 9 = (1000000/1010000) × 9 = 8.9 VThe output across the LDR is low in daylight and high in darkness.
Control Circuits and Automatic Systems
Potential dividers with sensors can provide input voltages to control circuits, enabling automatic switching based on environmental conditions.
Applications
| Application | Sensor | Circuit Design |
|---|---|---|
| Thermostat (heating) | Thermistor | High output when cold → turns heating ON |
| Refrigerator thermostat | Thermistor | High output when warm → turns cooling ON |
| Automatic street lights | LDR | High output in darkness → turns lights ON |
| Sunrise alarm | LDR | High output in light → triggers alarm |
To create a circuit that gives a HIGH output under certain conditions, place the sensor so that its resistance is LOW under those conditions relative to the fixed resistor, and take the output across the fixed resistor.
Variable Resistors for Adjustable Thresholds
Replacing the fixed resistor with a variable resistor allows the trigger point (threshold) to be adjusted. For example, in a thermostat, users can set their preferred temperature by adjusting the variable resistor.
Fixed vs Variable ResistorsWhy does an automatic car headlamp circuit use a fixed resistor, while an air-conditioning thermostat uses a variable resistor?
The headlamps always need to come on at the same low light level - there is no need for user adjustment. However, different people are comfortable at different temperatures, so the variable resistor enables them to set the room temperature to higher or lower values according to their preference.
Potential Divider as a Volume Control
FIG 10.14: Volume Control CircuitShow a potential divider with a +12 V supply at the top, a fixed resistor R₁ connected to a variable resistor R₂, with V_out taken from the junction between them. The bottom of R₂ connects to 0 V. Label that as R₂ increases, V_out increases.
In a volume control, the variable resistor is adjusted to change the output voltage that feeds into an amplifier. Increasing R_2 increases the fraction (R_2/R_1 + R_2) and hence increases V_out, making the sound louder.
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