AS Level Physics (9702): Comprehensive Study Notes on D.C. Circuits (Topic 10)

Welcome to the world of D.C. Circuits! This chapter is where the concepts of current, voltage, and resistance come together to help us analyze real-world electrical networks. Don't worry if complex circuits seem intimidating—we will break down the fundamental rules (Kirchhoff's Laws) into simple, manageable steps. Mastering this topic is essential for both your exam success and understanding how electronics work!

Remember: D.C. stands for Direct Current, meaning the current flows consistently in one direction.

10.1 Practical Circuits and Sources of E.M.F.

Circuit Symbols

In Physics 9702, you must be able to recall, draw, and interpret standard circuit diagrams using accepted symbols. These symbols include sources (cell, battery, power supply), resistors (fixed, variable, LDR, thermistor), switches, meters (ammeter, voltmeter), and components like the galvanometer and diode.

Tip: If you struggle with drawing circuits, practice sketching them quickly and neatly, ensuring all connections are clear and straight.

Electromotive Force (e.m.f.) vs. Potential Difference (p.d.)

This is arguably the most common conceptual mistake AS students make. Let's clarify the difference:

1. Potential Difference (p.d.), \(V\)

  • Definition: P.D. is the energy transferred or converted from electrical energy to other forms (like heat or light) per unit charge, when the charge passes through a component (e.g., a resistor or lamp).
  • Formula: \(V = \frac{W}{Q}\) (where \(W\) is energy transferred, \(Q\) is charge).
  • Analogy: This is the energy lost or "dropped" by the charge as it passes through a device.

2. Electromotive Force (e.m.f.), \(E\)

  • Definition: E.M.F. is the energy transferred from chemical or other forms to electrical energy per unit charge, in driving charge around a complete circuit (when passing through the source).
  • Formula: \(E = \frac{W_{\text{source}}}{Q}\).
  • Analogy: This is the total energy or "push" provided by the battery or power supply. Think of it as a water pump supplying energy to lift the water (charge).
Internal Resistance (\(r\))

All real sources of e.m.f. (batteries, generators) have some resistance within themselves. This is called internal resistance (\(r\)).

  • When charge flows through the source, some electrical energy is wasted (usually as heat) due to this internal resistance.
  • This wasted energy means the actual voltage available to the external circuit is less than the theoretical e.m.f.

The voltage across the external load resistance (\(R\)) is called the Terminal Potential Difference (\(V\)).

The e.m.f. must equal the total P.D. in the circuit (conservation of energy):
$$ E = V_{\text{external}} + V_{\text{internal}} $$

Using Ohm's law (\(V=IR\)) for the entire circuit where \(R_{\text{total}} = R + r\):

$$ E = I(R + r) $$

Since \(V\) is the P.D. across the external load \(R\) (so \(V = IR\)), we can rewrite the equation:

$$ E = V + Ir $$ $$ V = E - Ir $$

Key Point: The terminal P.D. (\(V\)) is always less than the e.m.f. (\(E\)) when current (\(I\)) is flowing, because of the voltage drop (\(Ir\)) across the internal resistance.

If the circuit is open (no current flows, \(I=0\)), then \(V = E\). This is how you measure the e.m.f. of a cell.

Quick Review:
| Term | Energy Transfer | Measurement | | :--- | :--- | :--- | | E.M.F. (E) | Chemical to Electrical | Total energy supplied by source (only measurable when $I=0$) | | Terminal P.D. (V) | Electrical to External Load | $V = E - Ir$ |


10.2 Kirchhoff's Laws: The Rules of Circuits

Kirchhoff's Laws are essential for analyzing complex circuits that cannot be solved using simple series/parallel rules.

Kirchhoff's First Law (The Current Law)

Statement: The sum of currents entering a junction is equal to the sum of currents leaving that junction.

$$ \Sigma I_{\text{in}} = \Sigma I_{\text{out}} $$

Underlying Principle: Conservation of Charge. Since charge cannot be created or destroyed, any charge flowing into a point must flow out of that point immediately.

Analogy: Imagine a water pipe junction. The total volume of water flowing in must equal the total volume of water flowing out.

Kirchhoff's Second Law (The Voltage Law)

Statement: In any closed loop in a circuit, the algebraic sum of the e.m.f.s is equal to the algebraic sum of the potential differences (voltage drops).

$$ \Sigma E = \Sigma IR $$

Underlying Principle: Conservation of Energy. When a unit of charge completes a full loop, the total energy supplied to it by sources (e.m.f.s) must equal the total energy taken from it by components (p.d.s).

Analogy: A rollercoaster starts and ends at the same height. Any energy (height) gained from the lift hill (E.M.F.) must be exactly used up by the friction and track features (P.D.s) throughout the ride.

How to Use Kirchhoff's Laws to Solve Problems:

When solving complex networks, you typically:

  1. Assign currents (\(I_1, I_2, I_3\), etc.) to every segment, choosing a direction arbitrarily (if you get a negative answer, the actual direction is opposite).
  2. Apply Kirchhoff I at junctions to relate the currents.
  3. Apply Kirchhoff II to all independent loops (closed paths) to generate voltage equations.
  4. Solve the simultaneous equations created in steps 2 and 3.

Don't worry if this seems tricky at first. Practice circuit problems systematically, and the method will become second nature.

Combining Resistors (Derivations using Kirchhoff's Laws)

You need to be able to derive and use the formulae for series and parallel resistance combinations, and understand that these derivations are based on Kirchhoff's Laws.

1. Resistors in Series

  • Rule: The current (\(I\)) is the same through all resistors.
  • Kirchhoff II Application: The total P.D. (\(V_T\)) is the sum of the P.D.s across individual resistors: \(V_T = V_1 + V_2\).
  • Derivation: $$ IR_T = IR_1 + IR_2 $$ (Divide by \(I\)) $$ R_T = R_1 + R_2 $$
  • Formula (Series): \(R_{\text{series}} = R_1 + R_2 + \dots\)

2. Resistors in Parallel

  • Rule: The P.D. (\(V\)) across all resistors is the same.
  • Kirchhoff I Application: The total current (\(I_T\)) splits, so \(I_T = I_1 + I_2\).
  • Derivation: Using \(I = V/R\): $$ \frac{V}{R_T} = \frac{V}{R_1} + \frac{V}{R_2} $$ (Divide by \(V\)) $$ \frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} $$
  • Formula (Parallel): \(\frac{1}{R_{\text{parallel}}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots\)

Key Takeaway for Kirchhoff's Laws: Kirchhoff I links to charge conservation (currents at a junction). Kirchhoff II links to energy conservation (voltages in a loop).

10.3 Potential Dividers

The Potential Divider Principle

A potential divider is a circuit consisting of resistors connected in series across a voltage source. Its purpose is to divide the source voltage into smaller, usable voltages.

Consider two resistors, \(R_1\) and \(R_2\), in series with an input voltage \(V_{\text{in}}\). The current \(I\) is constant:

$$ I = \frac{V_{\text{in}}}{R_1 + R_2} $$

If the output voltage (\(V_{\text{out}}\)) is taken across \(R_2\), then \(V_{\text{out}} = IR_2\). Substituting the current \(I\):

$$ V_{\text{out}} = V_{\text{in}} \times \frac{R_2}{R_1 + R_2} $$

The ratio of the output P.D. to the input P.D. is the same as the ratio of the output resistance (\(R_2\)) to the total resistance (\(R_1 + R_2\)).

The Potentiometer (Variable Potential Divider)

A potentiometer is a device used to provide a smoothly variable output voltage, or, more accurately, used to compare potential differences using a null method.

  • Structure: It consists of a uniform resistance wire of length \(L\) connected across a source. A sliding contact (jockey) allows you to tap off the voltage across a variable length \(x\).
  • Operation: Since resistance \(R\) is proportional to length \(L\) (\(R \propto L\)), the potential difference \(V\) across any length \(x\) is proportional to that length.
  • Comparing P.D.s: In a null method, a galvanometer is used to find the balance point where the voltage tapped off the potentiometer wire exactly equals the unknown P.D. (or E.M.F.). At this point, the galvanometer reads zero current (hence "null").

Did you know? Because the null method draws zero current from the source being measured, it avoids the errors introduced by internal resistance, making it highly accurate for P.D. comparisons.

Thermistors and LDRs in Potential Dividers

Thermistors and Light-Dependent Resistors (LDRs) are sensing devices whose resistance changes with external physical conditions (temperature and light intensity, respectively).

When placed into a potential divider circuit, they allow the output P.D. (\(V_{\text{out}}\)) to become dependent on the physical condition, making them useful in sensor circuits (e.g., thermostats or automatic lighting).

1. Light-Dependent Resistor (LDR)

  • Property: Resistance decreases as light intensity increases.
  • Application: If the LDR is \(R_2\), as light increases, \(R_2\) decreases. Since \(R_2\) is in the numerator of the divider equation, \(V_{\text{out}}\) decreases. (Useful for turning off lights when it gets bright.)

2. Thermistor (NTC type)

  • Property: Resistance decreases as temperature increases (Negative Temperature Coefficient - NTC).
  • Application: If the Thermistor is \(R_2\), as temperature increases, \(R_2\) decreases, and thus \(V_{\text{out}}\) decreases. (Useful for activating a fan when the temperature gets too high, or an alarm when the temperature drops too low.)

Quick Review: Potential Dividers

The potential divider is a voltage control device. When using a sensor (like an LDR or thermistor):

  • The resistance change in the sensor directly controls the output voltage ratio.
  • We choose the fixed resistor based on whether we want \(V_{\text{out}}\) to increase or decrease when the physical condition changes.

Congratulations on completing the study of D.C. Circuits! Remember the core ideas: energy conservation (Kirchhoff II, E.M.F. vs P.D.) and charge conservation (Kirchhoff I). Keep practicing those circuit problem-solving skills!