IP Physics Notes (Upper Secondary, Year 3-4): 12) Direct Current Circuits

Quick recap -- DC circuit analysis hinges on current and voltage sharing rules. Once you master how resistances combine and how potential dividers behave, thermistor/LDR sensor circuits become straightforward.

Series vs Parallel Essentials

• Series
  • Current is identical everywhere in the loop.
  • Potential differences add: \( V = V_1 + V_2 + \cdots \).
  • Resistances add: \( R_\text{eq} = R_1 + R_2 + \cdots \).
• Parallel
  • Potential difference across each branch is the same.
  • Currents add: \( I = I_1 + I_2 + \cdots \).
  • Inverse resistances add: \( 1 / R_\text{eq} = 1 / R_1 + 1 / R_2 + \cdots \).
• Power expressions: \( P = V I = I^2 R = \dfrac{V^2}{R} \).
• Ammeter -> series (assume \( R \approx 0 \)); voltmeter -> parallel (assume \( R \to \infty \)).

Worked Example: Mixed Network

A \( \pu{12 V} \) supply feeds a \( \pu{4 \Omega} \) resistor in series with a parallel pair of \( \pu{6 \Omega} \) and \( \pu{3 \Omega} \). The combined parallel resistance \( R_\text{parallel} = (1/6 + 1/3)^{-1} = \pu{2 \Omega} \), so total \( R_\text{eq} = 4 + 2 = \pu{6 \Omega} \). Total current \( I = 12 / 6 = \pu{2.0 A} \). Voltage across the parallel loop is \( V = I \times 2 = \pu{4.0 V} \), so branch currents are \( 4/6 = \pu{0.67 A} \) and \( 4/3 = \pu{1.33 A} \), summing back to \( 2.0 \) as a consistency check.

Potential Divider Principle

• Two series resistors \( R_1 \) and \( R_2 \) share supply \( V_\text{in} \).
• Output across \( R_2 \): \[ V_\text{out} = \frac{R_2}{R_1 + R_2} V_\text{in} \]
• Replace one resistor with a sensor (thermistor or LDR) to make the output voltage depend on temperature or light.

Sensor Behaviours

• NTC thermistor: resistance decreases as temperature rises. In a divider, place it top or bottom depending on whether you want output to increase or decrease with temperature.
• LDR: resistance decreases under intense light.
• Potentiometer: a three-terminal variable resistor acting as an adjustable divider (volume knobs, contrast controls).

Tips for Divider Problems

1. Identify which resistor the output spans.
2. Express \( V_\text{out} \) symbolically using the divider formula.
3. Substitute the sensor's resistance at the specified condition.
4. For switching circuits, combine with transistor/relay thresholds or compare to reference voltages.

Key Takeaways

• Memorise the sharing rules and reciprocal resistance formula for rapid simplification.
• Track total current first, then drop across series components to analyse parallel branches.
• Potential dividers turn resistance changes into voltage signals -- crucial for sensor questions.
• Always cross-check current splits and voltage sums to guard against arithmetic slips.