A-Level Physics D.C. Circuits Complete Guide

Core Concepts Practical Circuits Worked Examples Common Pitfalls Quick Reference

            🎯 Exam Focus: D.C. Circuits appears in 5-8 questions per exam. Master this topic for 10-15% of total marks!
        

1. Core Concepts

e.m.f. vs. Potential Difference

Term	Definition	Formula	Energy
e.m.f. (E)	Total energy per unit charge from source	E = Energy/Charge	Chemical → Electrical
p.d. (V)	Energy per unit charge transferred	V = Work/Charge	Electrical → Heat/Light

Internal Resistance

V = E - Ir

Key point: Terminal p.d. < e.m.f. when current flows.

Kirchhoff's Laws

First Law: ΣI_in = ΣI_out (Conservation of charge)

Second Law: ΣE = Σ(IR) (Conservation of energy)

Resistor Combinations

Series: R_total = R₁ + R₂ + R₃ + ...

Parallel: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + ...

2. Practical Circuits

Potential Dividers

V_out = [R₂/(R₁ + R₂)] × V_in

Used for: Variable voltage supplies, sensor circuits.

Potentiometers

Null method – compares e.m.f.s without drawing current.

E₁/E₂ = l₁/l₂

Sensor Circuits

Sensor	Resistance Change	Application
Thermistor (NTC)	↓ as temperature ↑	Temperature alarms
LDR	↓ as light intensity ↑	Light sensors

3. Worked Examples

Example 1: Internal Resistance

Problem: Battery (e.m.f. 9.0V) connected to 4.0Ω resistor. Power = 8.0W. Find r.

Solution:

1. P = I²R → 8.0 = I² × 4.0 → I = √2 = 1.414A

2. V = IR = 1.414 × 4.0 = 5.656V

3. V = E - Ir → 5.656 = 9.0 - 1.414r

4. r = 2.37Ω

Example 2: Potentiometer

Problem: Balance length = 41.0cm for 1.23V. What series R gives balance at 50.0cm?

Solution:

1. Initial p.d./length = 1.23/0.410 = 3.00 V/m

2. Required = 1.23/0.500 = 2.46 V/m

3. (3.0 × 4.0)/(4.0 + R_s) = 2.46

4. R_s = 0.88Ω

4. Common Pitfalls

⚠️ Watch Out!
Signs in Kirchhoff's 2nd Law: Most common error! IR drop is negative if traversing with current.

💡 Pro Tip
Maximum Power: External R gets max power when R = internal resistance r.

📈 Graph Skills
V = E - Ir gives straight line: intercept = E, gradient = -r

📋 Quick Reference

Internal Resistance: V = E - Ir

Kirchhoff 1st: ΣI_in = ΣI_out

Kirchhoff 2nd: ΣE = Σ(IR)

Series: R_T = R₁ + R₂ + ...

Parallel: 1/R_T = 1/R₁ + 1/R₂ + ...

Power: P = VI = I²R = V²/R

Potential Divider: V_out = [R₂/(R₁+R₂)]V_in

Potentiometer: E₁/E₂ = l₁/l₂

📚 Based on CIE A-Level Physics 9702 Syllabus

Complete A-Level Physics D.C. Circuits Guide - Syllabus Coverage

CIE A-Level Physics 9702: Complete D.C. Circuits Guide

Syllabus Section 10 | Paper 1, 2, & 4 | 2023-2025 Syllabus

10.1

Practical Circuits

Learning Objectives

Recall and use standard circuit symbols

Draw and interpret circuit diagrams

Define e.m.f. as energy transferred per unit charge

Distinguish between e.m.f. and p.d. in terms of energy

Understand effects of internal resistance on terminal p.d.

Electromotive Force (e.m.f.) - Formal Definition

e.m.f. (E) = The total energy transferred per unit charge from a source in driving charge around a complete circuit.

E = W / Q where W = total energy transferred from source (J), Q = charge (C)

Physical meaning: When 1 coulomb of charge passes through the source, it gains E joules of electrical energy from chemical/other forms.

Potential Difference (p.d.) - Formal Definition

p.d. (V) = The energy transferred per unit charge from electrical to other forms when charge passes between two points.

V = W / Q where W = work done/energy dissipated (J), Q = charge (C)

Physical meaning: When 1 coulomb passes between two points, it transfers V joules to heat/light/other forms.

Aspect	e.m.f. (E)	Potential Difference (V)
Definition	Energy supplied per unit charge by source	Energy transferred per unit charge between two points
Circuit Role	Cause of current flow (driving force)	Effect of current flow (energy use)
Measurement	Measured when source supplies no current (open circuit)	Measured when current is flowing
Energy Change	Non-electrical → Electrical	Electrical → Other forms
Symbol in Equations	E or ε	V or ΔV

Exam Application: Internal Resistance

Syllabus requirement: "Understand the effects of the internal resistance of a source of e.m.f. on the terminal potential difference"

V = E - Ir

Where:
V = terminal p.d. (voltage across source terminals)
E = e.m.f. of source
I = current in circuit
r = internal resistance

Step 1: When switch is open (I=0), terminal p.d. = e.m.f.

Step 2: When current flows, energy is dissipated inside source as I²r

Step 3: Terminal p.d. is less than e.m.f. by amount Ir

Step 4: Maximum current occurs when terminals are short-circuited: I_max = E/r

10.2

Kirchhoff's Laws

Learning Objectives

Recall Kirchhoff's first law and understand conservation of charge basis

Recall Kirchhoff's second law and understand conservation of energy basis

Derive formulas for series and parallel resistors using Kirchhoff's laws

Use formulas for combined resistance

Solve circuit problems using Kirchhoff's laws

Kirchhoff's First Law (Junction Rule)

Statement: The sum of currents entering any junction equals the sum of currents leaving that junction.

ΣI_in = ΣI_out

Conservation Basis: Charge cannot be created or destroyed at a junction. This is a direct consequence of conservation of charge.

Mathematical form: Taking currents entering as positive and leaving as negative: ΣI = 0 at any junction.

Kirchhoff's Second Law (Loop Rule)

Statement: The sum of e.m.f.s in any closed loop equals the sum of potential differences in that loop.

ΣE = ΣV = Σ(IR)

Conservation Basis: Energy supplied by sources equals energy dissipated in components. This is a direct consequence of conservation of energy.

Sign Convention Critical: Choose consistent direction for each loop. E.m.f. is positive if traversed from - to + terminal. IR drop is negative if traversed in same direction as current.

Derivation: Resistors in Series (Syllabus Requirement)

Given: Resistors R₁, R₂, R₃ in series with current I and total p.d. V

Apply Kirchhoff's 2nd law to the loop: V = V₁ + V₂ + V₃

Using Ohm's law for each resistor: V = IR₁ + IR₂ + IR₃

Factor out I: V = I(R₁ + R₂ + R₃)

But V = IR_eq where R_eq is equivalent resistance

Therefore: IR_eq = I(R₁ + R₂ + R₃)

Cancel I: R_eq = R₁ + R₂ + R₃

R_series = R₁ + R₂ + R₃ + ...

Derivation: Resistors in Parallel (Syllabus Requirement)

Given: Resistors R₁, R₂, R₃ in parallel with total current I and p.d. V across each

Apply Kirchhoff's 1st law at junction: I = I₁ + I₂ + I₃

Using Ohm's law for each branch: I = V/R₁ + V/R₂ + V/R₃

Factor out V: I = V(1/R₁ + 1/R₂ + 1/R₃)

But I = V/R_eq where R_eq is equivalent resistance

Therefore: V/R_eq = V(1/R₁ + 1/R₂ + 1/R₃)

Cancel V: 1/R_eq = 1/R₁ + 1/R₂ + 1/R₃

1/R_parallel = 1/R₁ + 1/R₂ + 1/R₃ + ...

Special case for two resistors: R_eq = (R₁R₂)/(R₁ + R₂)

Circuit Problem Solving Strategy (Syllabus Requirement)

Step-by-step approach for complex circuits:

Step 1: Label all known and unknown quantities

Step 2: Assign current directions (guess if unknown)

Step 3: Apply Kirchhoff's 1st law at all junctions

Step 4: Apply Kirchhoff's 2nd law to enough independent loops

Step 5: Solve simultaneous equations

Step 6: If any current is negative, reverse its direction in your diagram

Example circuit types to practice:

Multiple battery circuits
Bridge circuits
Networks with mixed series-parallel combinations
Circuits with internal resistance

10.3

Potential Dividers

Learning Objectives

Understand the principle of a potential divider circuit

Recall and use the potentiometer principle for comparing p.d.s

Understand use of galvanometers in null methods

Explain use of thermistors and LDRs in potential dividers

Potential Divider Principle

Definition: A circuit that divides the input voltage into smaller output voltages using resistors.

V_out = [R₂/(R₁ + R₂)] × V_in

Derivation from Ohm's Law:

Current in circuit: I = V_in/(R₁ + R₂)

Output voltage across R₂: V_out = I × R₂

Substitute I: V_out = [V_in/(R₁ + R₂)] × R₂

Simplify: V_out = V_in × [R₂/(R₁ + R₂)]

Key features:

Output is always less than input
Output proportional to resistance ratio
Can provide variable output if one resistor is variable

Potentiometer Principle

Definition: A null method device for accurately comparing e.m.f.s or measuring p.d.s without drawing current from the unknown source.

E_x/E_s = l_x/l_s

Why it's accurate (null method):

At balance point, galvanometer reads zero

No current flows through unknown cell

Therefore, internal resistance of unknown cell has no effect

Measurement depends only on length ratio and known e.m.f.

Requirements for accurate potentiometer:

Uniform resistance wire (constant resistance per unit length)
Driver cell e.m.f. > unknown e.m.f.
Sensitive galvanometer
Low resistance connecting wires

Galvanometer in Null Methods

Syllabus requirement: "Understand the use of a galvanometer in null methods"

Galvanometer characteristics:

Very sensitive current detector
Can detect current in either direction
Shows zero when p.d. across it is zero
Does not need calibration when used in null method

Advantages of null method:

No current drawn from unknown source during measurement
Internal resistance doesn't affect measurement
High accuracy as depends on length ratio
Galvanometer sensitivity more important than calibration

Thermistors in Potential Dividers

Resistance characteristic: NTC thermistors decrease resistance as temperature increases

Typical R-T relationship: R = R₀e^(β(1/T - 1/T₀))

Circuit configuration: Usually in top arm of potential divider

Output behavior: V_out increases as temperature increases

Applications: Temperature alarms, thermostats, fire alarms

Exam analysis: For fixed R in lower arm, V_out ∝ R_thermistor/(R_fixed + R_thermistor)

Light-Dependent Resistors (LDRs)

Resistance characteristic: Decreases as light intensity increases

Typical range: MΩ in dark → kΩ in bright light

Circuit configuration: Position depends on required output

Output behavior: V_out increases with light if LDR in lower arm

Applications: Light meters, automatic lighting, security systems

Exam analysis: Often combined with comparators/transistors to trigger at threshold

Designing Sensor Circuits (Syllabus Requirement)

Step-by-step design process:

Step 1: Determine resistance range of sensor

Step 2: Choose fixed resistor to give desired output range

Step 3: Calculate V_out(min) and V_out(max)

Step 4: Position sensor in appropriate arm of divider

Step 5: Add comparator/transistor if switching action needed

Example exam question structure:

"The resistance of the thermistor is 2000Ω at 20°C and 500Ω at 60°C. Design a circuit to give 0V output at 20°C and 5V output at 60°C."
"The LDR has resistance 100kΩ in dark and 1kΩ in light. Choose a fixed resistor to give maximum voltage change."

Syllabus Coverage Complete

All learning objectives from Section 10 (D.C. Circuits) of CIE A-Level Physics 9702 syllabus addressed

Paper 1: Multiple choice applications | Paper 2: Structured questions | Paper 4: Data analysis & planning

```

Circuit Symbols: Interpretation and Function

Cell

Interpretation: A single electrical energy source

Function: Supplies electrical energy to a circuit

Battery of cells

Interpretation: Two or more cells connected together

Function: Provides a higher potential difference

Power supply

Interpretation: An external electrical energy source

Function: Supplies electrical energy to a circuit

a.c. power supply

Interpretation: An alternating current source

Function: Supplies alternating voltage and current

Earth

Interpretation: Zero-potential reference point

Function: Provides a common reference potential

Switch

Interpretation: A control device

Function: Opens or closes a circuit

Junction of conductors

Interpretation: A connection point of wires

Function: Allows current to split or combine

Lamp

Interpretation: A light-producing component

Function: Converts electrical energy to light

Heater

Interpretation: A heating component

Function: Converts electrical energy to heat

Motor (M)

Interpretation: An electric motor

Function: Converts electrical energy to kinetic energy

Generator (G)

Interpretation: An electrical generator

Function: Converts kinetic energy to electrical energy

Electric bell

Interpretation: A sound-producing device

Function: Produces sound when current flows

Buzzer

Interpretation: An audible signalling device

Function: Produces sound when current flows

Loudspeaker

Interpretation: A sound output device

Function: Converts electrical signals into sound

Microphone

Interpretation: A sound input device

Function: Converts sound into electrical signals

Fixed resistor

Interpretation: A resistor with constant resistance

Function: Limits current in a circuit

Variable resistor

Interpretation: An adjustable resistor

Function: Used to vary current or potential difference

Potentiometer

Interpretation: A variable resistor with three terminals

Function: Used as a potential divider

Thermistor

Interpretation: A temperature-dependent resistor

Function: Changes resistance with temperature

Light-dependent resistor (LDR)

Interpretation: A light-sensitive resistor

Function: Changes resistance with light intensity

Ammeter (A)

Interpretation: A current-measuring instrument

Function: Measures electric current

Voltmeter (V)

Interpretation: A potential difference measuring instrument

Function: Measures voltage across components

Galvanometer

Interpretation: A sensitive current detector

Function: Detects small electric currents

Oscilloscope

Interpretation: A voltage display instrument

Function: Displays voltage variation with time

Diode

Interpretation: A one-directional device

Function: Allows current to flow in one direction only

Light-emitting diode (LED)

Interpretation: A light-producing diode

Function: Emits light when current flows

Capacitor

Interpretation: A charge-storing component

Function: Stores electrical charge

📚 Master A-Level Physics D.C. Circuits

1. Core Concepts

e.m.f. vs. Potential Difference

Internal Resistance

Kirchhoff's Laws

Resistor Combinations

2. Practical Circuits

Potential Dividers

Potentiometers

Sensor Circuits

3. Worked Examples

Example 1: Internal Resistance

Example 2: Potentiometer

4. Common Pitfalls

📋 Quick Reference

CIE A-Level Physics 9702: Complete D.C. Circuits Guide

Learning Objectives

Electromotive Force (e.m.f.) - Formal Definition

Potential Difference (p.d.) - Formal Definition

Exam Application: Internal Resistance

Learning Objectives

Kirchhoff's First Law (Junction Rule)

Kirchhoff's Second Law (Loop Rule)

Derivation: Resistors in Series (Syllabus Requirement)

Derivation: Resistors in Parallel (Syllabus Requirement)

Circuit Problem Solving Strategy (Syllabus Requirement)

Learning Objectives

Potential Divider Principle

Potentiometer Principle

Galvanometer in Null Methods

Thermistors in Potential Dividers

Light-Dependent Resistors (LDRs)

Designing Sensor Circuits (Syllabus Requirement)

Syllabus Coverage Complete

Circuit Symbols: Interpretation and Function

Cell

Battery of cells

Power supply

a.c. power supply

Earth

Switch

Junction of conductors

Lamp

Heater

Motor (M)

Generator (G)

Electric bell

Buzzer

Loudspeaker

Microphone

Fixed resistor

Variable resistor

Potentiometer

Thermistor

Light-dependent resistor (LDR)

Ammeter (A)

Voltmeter (V)

Galvanometer

Oscilloscope

Diode

Light-emitting diode (LED)

Capacitor

As Physics Topic 10:D.C. circuits