Chapter 1.12: Potential Divider Circuits
Welcome to one of the most practical and frequently examined topics in AS 1: Forces, Energy and Electricity! Have you ever wondered how a dimmer switch dims a light bulb, how a smartphone adjusts its screen brightness in the dark, or how an oven maintains a set temperature? All of these devices rely on potential divider circuits.
Don't worry if circuit calculations have felt daunting before. A potential divider is simply a circuit that shares out voltage. Once you master a few straightforward rules, you will be able to solve these exam questions with complete confidence.
---1. What is a Potential Divider?
A potential divider is a simple circuit consisting of two or more resistors (or components) connected in series across a voltage supply (\(V_{\text{in}}\)). Its job is to split (divide) the total input voltage into smaller, useful output voltages (\(V_{\text{out}}\)).
The Core Principle: Voltage Sharing
In a series circuit:
• The same electric current \(I\) flows through every component.
• The total resistance is the sum of the individual resistances: \(R_{\text{total}} = R_1 + R_2\).
• The total current flowing through the circuit is given by Ohm's law: \(I = \frac{V_{\text{in}}}{R_1 + R_2}\).
Since the potential difference across any resistor is \(V = IR\), and the current \(I\) is constant throughout, the potential difference across a resistor is directly proportional to its resistance (\(V \propto R\)).
The Golden Rule: The bigger the resistance, the bigger its share of the voltage!
---2. The Essential Mathematical Formulae
The Standard Potential Divider Equation
If two resistors, \(R_1\) and \(R_2\), are connected in series across an input voltage \(V_{\text{in}}\), and we take the output voltage \(V_{\text{out}}\) across \(R_2\), the relationship is:
\(V_{\text{out}} = \left( \frac{R_2}{R_1 + R_2} \right) V_{\text{in}}\)
Notice what this equation is doing: the fraction \(\frac{R_2}{R_1 + R_2}\) is simply the fraction of the total resistance that belongs to \(R_2\). Multiplying this fraction by \(V_{\text{in}}\) gives the exact voltage across \(R_2\).
The Ratio Form
You can also compare the voltages across individual resistors directly using ratios:
\(\frac{V_1}{V_2} = \frac{R_1}{R_2}\)
\(\frac{V_1}{V_{\text{total}}} = \frac{R_1}{R_1 + R_2}\)
Step-by-Step Worked Example
Question: A \(12\text{ V}\) supply is connected across two series resistors: \(R_1 = 40\,\Omega\) and \(R_2 = 60\,\Omega\). Calculate the output voltage \(V_{\text{out}}\) measured across \(R_2\).
Step 1: Identify the known values.
\(V_{\text{in}} = 12\text{ V}\), \(R_1 = 40\,\Omega\), \(R_2 = 60\,\Omega\).
Step 2: Calculate total resistance.
\(R_{\text{total}} = R_1 + R_2 = 40\,\Omega + 60\,\Omega = 100\,\Omega\).
Step 3: Apply the potential divider formula.
\(V_{\text{out}} = \left( \frac{60}{40 + 60} \right) \times 12\text{ V} = \left( \frac{60}{100} \right) \times 12\text{ V} = 0.60 \times 12\text{ V} = 7.2\text{ V}\).
Quick Check: The voltage across \(R_1\) must be \(12\text{ V} - 7.2\text{ V} = 4.8\text{ V}\). Since \(60\,\Omega > 40\,\Omega\), \(R_2\) receives the larger share of the voltage, which matches our calculation.
---3. Continuous Potentiometers (Resistance Wires)
Instead of two separate fixed resistors, a potential divider can be built using a single uniform length of resistance wire with a sliding contact (a potentiometer).
Because the wire is uniform, resistance is directly proportional to length (\(R \propto l\)). If a wire has a total length \(L\) and you tap off an output across a length \(l\):
\(V_{\text{out}} = \left( \frac{l}{L} \right) V_{\text{in}}\)
Why is this useful? By sliding the contact smoothly from \(l = 0\) to \(l = L\), you can adjust the output voltage continuously from \(0\text{ V}\) all the way up to \(V_{\text{in}}\). This allows complete control over an output component, such as dimming a lamp down to zero brightness.
---4. Sensing and Control Circuits
Potential dividers become extremely powerful when we replace one of the fixed resistors with an environmental sensor, such as an LDR or an NTC thermistor.
1. Light-Dependent Resistor (LDR)
• Behavior: As light intensity increases, its resistance decreases.
• Memory Trick: LURD — Light Up \(\rightarrow\) Resistance Down.
• Application: Automatic street lighting and darkness/daylight detectors.
2. Negative Temperature Coefficient (NTC) Thermistor
• Behavior: As temperature increases, its resistance decreases.
• Memory Trick: TURD — Temperature Up \(\rightarrow\) Resistance Down.
• Application: Digital thermometers, frost alarms, thermostat heaters, and fire alarms.
3. How Sensor Circuits Work in 3 Logical Steps
When explaining how a sensor circuit responds in an exam, always follow this 3-step chain of reasoning:
1. Condition Change: State what happens to the sensor's resistance (e.g. "When temperature rises, the thermistor's resistance decreases").
2. Voltage Across Sensor: Because its resistance drops, the sensor takes a smaller fraction of the total voltage (its p.d. decreases).
3. Voltage Across Partner Resistor: Because the total supply voltage \(V_{\text{in}}\) is constant (\(V_{\text{in}} = V_{\text{sensor}} + V_{\text{fixed}}\)), the potential difference across the fixed resistor must increase.
Quick Example: Designing a Street Light Trigger
We want an output voltage that increases when it gets dark to trigger a lamp switch.
• In the dark, the LDR's resistance increases.
• The LDR now claims a larger share of the input voltage.
• Therefore, we must place our output connections (\(V_{\text{out}}\)) across the LDR!
5. Circuit Loading and Internal Resistance
The Effect of Load Resistance
In theory, we calculate \(V_{\text{out}}\) assuming no current leaves the divider. However, in real circuits, connecting an external component (a "load" \(R_L\)) or a real voltmeter across \(R_2\) alters the circuit:
• Ideal Voltmeter: Has infinite resistance (\(R_V \to \infty\)), draws zero current, and does not alter \(V_{\text{out}}\).
• Finite Load Resistor (\(R_L\)): When connected in parallel with \(R_2\), the combined resistance becomes:
\(R_p = \frac{R_2 R_L}{R_2 + R_L}\)
Because \(R_p\) is always smaller than \(R_2\) alone, the effective resistance of that branch drops. Consequently, the actual output voltage \(V_{\text{out}}\) decreases below its unloaded value.
Supply Internal Resistance
If the power supply has internal resistance (\(r\)), the voltage supplied to the divider is not the full EMF (\(\mathcal{E}\)), but the terminal potential difference:
\(V_{\text{in}} = \mathcal{E} - Ir\)
As current is drawn by the divider, "lost volts" (\(Ir\)) develop inside the cell, reducing the available \(V_{\text{in}}\).
---6. Common Pitfalls & Examiner Tips
• Inverting the Numerator: Always double-check which resistor is in the numerator! If you want \(V_{\text{out}}\) across \(R_2\), use \(R_2\) on top: \(V_{\text{out}} = \left( \frac{R_2}{R_1 + R_2} \right) V_{\text{in}}\).
• Opposite Sensor Actions: Remember that an NTC thermistor's resistance goes down when heated, not up.
• Forgetting the Series Partner: If a thermistor is heated, the voltage across the thermistor drops, but the voltage across the fixed resistor in series with it goes UP. Make sure you check which component \(V_{\text{out}}\) is connected across.
• Loaded Dividers: If an exam question mentions a load resistor connected across the output, always calculate the parallel equivalent resistance \(R_p\) first before calculating \(V_{\text{out}}\).
7. Key Takeaways
• A potential divider shares voltage in direct proportion to resistance (\(V \propto R\)).
• Standard equation: \(V_{\text{out}} = \left( \frac{R_2}{R_1 + R_2} \right) V_{\text{in}}\).
• Potentiometer wire equation: \(V_{\text{out}} = \left( \frac{l}{L} \right) V_{\text{in}}\).
• LDR: Light increases \(\rightarrow\) Resistance decreases.
• NTC Thermistor: Temperature increases \(\rightarrow\) Resistance decreases.
• Connecting a finite load in parallel across the output lowers the effective resistance and reduces \(V_{\text{out}}\).