Introduction to Potential Divider Circuits
Welcome to one of the most practical and exciting topics in AS Physics! Have you ever wondered how a dimmer switch dims a lightbulb, how a smartphone screen automatically adjusts its brightness in the dark, or how an oven knows when it has reached the right temperature? The answer lies in potential divider circuits.
Don't worry if electricity sometimes feels abstract or difficult to picture. In this chapter, we will break down how voltage splits across components, how sensors like thermistors and LDRs respond to their environment, and how to solve any potential divider calculation with ease.
1. What is a Potential Divider?
A potential divider (often called a voltage divider) is simply a series circuit consisting of two or more resistors connected across a voltage source. Its purpose is to "divide" or share the total supply voltage (potential difference) between the components.
How Voltage Splits in Series
Let's recap two golden rules of series circuits:
1. The current \(I\) is the same through all series components.
2. The total supply voltage \(V_{\text{in}}\) is shared between the resistors: \(V_{\text{in}} = V_1 + V_2\).
From Ohm's Law, \(V = I \times R\). Because the current \(I\) is identical through both resistors, the voltage across each resistor is directly proportional to its resistance. In simple terms: the bigger the resistance, the bigger its share of the total voltage!
The Potential Divider Equations
There are two handy ways to calculate the voltage across a resistor in a potential divider:
Method 1: The Ratio Method
The ratio of voltages equals the ratio of resistances:
\(\frac{V_1}{V_2} = \frac{R_1}{R_2}\)
Method 2: The Potential Divider Formula
If two resistors \(R_1\) and \(R_2\) are connected in series across an input voltage \(V_{\text{in}}\), and we take our output voltage \(V_{\text{out}}\) across \(R_2\), the formula is:
\(V_{\text{out}} = \left( \frac{R_2}{R_1 + R_2} \right) \times V_{\text{in}}\)
Memory Tip: To find the voltage across a specific resistor, put that resistor's value on top (numerator) and the total resistance on the bottom (denominator), then multiply by the total supply voltage.
Step-by-Step Worked Example
Question: A \(12\text{ V}\) battery is connected across two resistors in series: \(R_1 = 400\ \Omega\) and \(R_2 = 800\ \Omega\). Calculate the output voltage \(V_{\text{out}}\) across \(R_2\).
Step 1: Identify the known values.
\(V_{\text{in}} = 12\text{ V}\)
\(R_1 = 400\ \Omega\)
\(R_2 = 800\ \Omega\)
Step 2: Calculate total resistance \(R_{\text{total}}\).
\(R_{\text{total}} = R_1 + R_2 = 400\ \Omega + 800\ \Omega = 1200\ \Omega\)
Step 3: Apply the potential divider formula.
\(V_{\text{out}} = \left( \frac{R_2}{R_1 + R_2} \right) \times V_{\text{in}}\)
\(V_{\text{out}} = \left( \frac{800}{1200} \right) \times 12\text{ V} = \frac{2}{3} \times 12\text{ V} = 8\text{ V}\)
Quick Check: Since \(R_2\) is twice as large as \(R_1\), it gets twice the voltage (\(8\text{ V}\) vs \(4\text{ V}\)). Together, \(8\text{ V} + 4\text{ V} = 12\text{ V}\). Everything balances perfectly!
Key Takeaway: Potential dividers share out the total voltage in proportion to resistance. A larger resistance always grabs a larger share of the potential difference.
2. Producing a Continuously Variable Voltage
Fixed resistors are great for giving a fixed output voltage, but what if you want to turn a volume knob smoothly from minimum to maximum? For this, we use a potentiometer (a variable potential divider).
How a Potentiometer Works
A potentiometer consists of a long resistive track with a movable sliding contact (called a wiper):
- The full track has resistance \(R_{\text{total}}\) connected across the input supply \(V_{\text{in}}\).
- Moving the slider changes the position where the output voltage \(V_{\text{out}}\) is tapped off.
- When the slider is at the very bottom (connected to \(0\text{ V}\)), the output voltage is \(V_{\text{out}} = 0\text{ V}\).
- When the slider is moved to the very top, the output voltage is \(V_{\text{out}} = V_{\text{in}}\).
- Moving the slider smoothly varies \(V_{\text{out}}\) continuously anywhere between \(0\text{ V}\) and \(V_{\text{in}}\).
Variable Resistor (Rheostat) vs. Potentiometer
A common point of confusion in exams is the difference between connecting a variable resistor in series versus as a potentiometer:
- Rheostat (2 terminals used in series): Varies total circuit resistance to control current. It cannot reduce the potential difference across a load all the way to zero.
- Potentiometer (3 terminals used): Acts as a variable potential divider to control voltage smoothly from \(0\text{ V}\) to maximum.
Key Takeaway: A potentiometer allows full, continuous control of output voltage from \(0\text{ V}\) right up to the maximum supply voltage.
3. Sensor Circuits: Thermistors and LDRs
By replacing one of the fixed resistors with an environmental sensor, a potential divider can detect changes in temperature or light levels. These are known as sensor circuits or transducer circuits.
Meet the Sensors
1. Light Dependent Resistor (LDR)
An LDR changes its resistance depending on light intensity.
- Bright light: High number of free charge carriers \(\implies\) Low resistance.
- Darkness: Few free charge carriers \(\implies\) High resistance.
Memory Trick: LURD — Light Up, Resistance Down!
2. Negative Temperature Coefficient (NTC) Thermistor
In the CCEA specification, thermistors are assumed to be NTC (negative temperature coefficient).
- High temperature (hot): Thermal energy releases more free charge carriers \(\implies\) Low resistance.
- Low temperature (cold): Fewer charge carriers available \(\implies\) High resistance.
Memory Trick: TURD — Temperature Up, Resistance Down!
Analyzing Sensor Circuits Step-by-Step
When analyzing sensor circuits, always follow this foolproof 4-step logic chain:
1. Change: What happens to the physical condition (light or temperature)?
2. Resistance: Does the sensor's resistance go UP or DOWN?
3. Share of Voltage: Does the sensor get a LARGER or SMALLER share of \(V_{\text{in}}\)?
4. Output Voltage: Look at where \(V_{\text{out}}\) is measured across to determine if \(V_{\text{out}}\) rises or falls.
Example A: Automatic Street Light (Darkness Detector)
Imagine an LDR (\(R_1\)) placed in the top position and a fixed resistor (\(R_2\)) at the bottom. We measure \(V_{\text{out}}\) across the fixed resistor \(R_2\).
- As it gets dark: Light intensity decreases \(\implies\) LDR resistance increases significantly.
- The LDR takes a much larger share of the supply voltage.
- Therefore, the remaining voltage across the fixed resistor \(R_2\) decreases (\(V_{\text{out}}\) falls).
- Alternative design: If we measure \(V_{\text{out}}\) across the LDR instead, \(V_{\text{out}}\) will rise in the dark, which can be used to trigger a switch to turn on street lamps!
Example B: Temperature Warning Alarm (Overheating Detector)
A circuit consists of a fixed resistor \(R_1\) at the top and an NTC thermistor \(R_{\text{th}}\) at the bottom. The output voltage \(V_{\text{out}}\) is taken across the fixed resistor \(R_1\).
- When temperature increases: Thermistor resistance \(R_{\text{th}}\) drops.
- Total circuit resistance falls, so the thermistor takes a smaller fraction of the voltage.
- Consequently, the voltage across the fixed resistor \(R_1\) increases (\(V_{\text{out}}\) rises).
- This higher output voltage can trigger a cooling fan or buzzer.
Key Takeaway: Sensor potential dividers convert a physical change (temperature or light) into a changing electrical voltage signal.
4. Summary of Common Mistakes to Avoid
Mistake 1: Forgetting that total current changes in sensor circuits.
When the resistance of a thermistor or LDR changes, the total resistance of the entire circuit changes, which means the total current also changes. Don't assume current stays constant!
Mistake 2: Mixing up \(R_1\) and \(R_2\) in the formula.
Always double-check which resistor is connected to the output terminals. The numerator in \(\left(\frac{R}{R_{\text{total}}}\right)\) must always be the specific component across which you are measuring \(V_{\text{out}}\).
Mistake 3: Mixing up sensor responses.
Remember the mnemonics LURD (Light Up, Resistance Down) and TURD (Temperature Up, Resistance Down). In standard AS Physics sensors, increasing the environmental input always lowers the component's resistance.
Quick Review: Essential Formulas & Rules
- Potential Divider Formula: \(V_{\text{out}} = \left( \frac{R_2}{R_1 + R_2} \right) \times V_{\text{in}}\)
- Voltage Ratio Rule: \(\frac{V_1}{V_2} = \frac{R_1}{R_2}\)
- LDR: Dark \(\implies\) High \(R\) | Bright \(\implies\) Low \(R\)
- NTC Thermistor: Cold \(\implies\) High \(R\) | Hot \(\implies\) Low \(R\)
- Potentiometer Output Range: Continuous output from \(0\text{ V}\) up to \(V_{\text{in}}\)