DC Circuits

Overview

This chapter applies ideas from Current Electricity Fundamentals to complete direct-current circuits containing cells, resistors, lamps, meters and combinations of components.

Main skills:

• analyse current and potential difference in circuits
• combine resistors in series and parallel
• solve mixed resistor networks
• compare lamp brightness using power
• apply junction and loop reasoning
• understand potential dividers
• understand potentiometer null method
• interpret faults in circuits

Core Ideas

DC-circuit analysis rests on a small set of ideas used repeatedly:

• current is conserved at junctions
• potential difference represents energy transferred per unit charge
• resistors share current or p.d. according to how they are connected
• power determines heating and brightness
• balanced potentials imply zero current between points

Core Quantities and Notation

Current

Rate of flow of charge:

$$I=\frac{Q}{t}$$

Current is usually treated as a signed scalar in circuit analysis. A chosen direction is assigned first.

Potential Difference

Energy transferred per unit charge:

$$V=\frac{W}{Q}$$

Resistance

$$R=\frac{V}{I}$$

For ohmic conductors:

$$V=IR$$

Power

$$P=IV$$

Also:

$$P=I^{2}R$$ $$P=\frac{V^2}{R}$$

See also Work, Energy and Power.

Exam Relevance

DC circuit questions are usually won or lost on structure, not algebra. Students need to recognise series and parallel sections correctly, assign currents and p.d.s locally, and avoid treating formulas as global shortcuts. The topic is also a bridge to potentiometers, internal resistance, and fault diagnosis, so conceptual clarity matters as much as computation.

Series Circuits

Components connected one after another with no branching.

Rules

Current

Same through all components:

$$I_1=I_2=I_3$$

Potential Difference

Supply voltage shared:

$$V_{\text{total}}=V_1+V_2+V_3$$

Total Resistance

$$R_{\text{total}}=R_1+R_2+R_3+\cdots$$

Consequence

Adding more resistors in series increases total resistance and usually reduces current.

Parallel Circuits

Components connected across the same two junctions.

Rules

Potential Difference

Same across each branch:

$$V_1=V_2=V_3$$

Current

Current splits at junctions:

$$I_{\text{total}}=I_1+I_2+I_3$$

Total Resistance

$$\frac{1}{R_{\text{total}}}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+\cdots$$

Consequence

Adding extra branches reduces total resistance.

Current Distribution and Junction Reasoning

At any junction:

$$\text{sum of currents entering}=\text{sum of currents leaving}$$

This is conservation of charge.

Example

If $3.0\text{ A}$ enters a junction and one branch carries $1.2\text{ A}$:

$$I_{\text{other}}=1.8\text{ A}$$

Potential Difference and Loop Reasoning

Around a complete loop, gains and drops in potential must balance.

Example

Single cell with two series resistors:

$$\mathcal{E}=V_1+V_2$$

This is the basis of many H2 circuit equations.

Mixed Resistor Networks

Many exam questions contain both series and parallel parts.

Method

1. Identify obvious series groups.
2. Identify obvious parallel groups.
3. Replace step-by-step with equivalent resistance.
4. Find total current.
5. Work backwards to obtain branch currents or voltages.

Common Trap

Two resistors are only parallel if both ends connect to the same two junctions.

Worked Example 1: Mixed Network

A $2\Omega$ resistor is in series with two parallel resistors of $3\Omega$ and $6\Omega$.

Step 1: Parallel Part

$$\frac{1}{R_p}=\frac{1}{3}+\frac{1}{6}=\frac{1}{2}$$ $$R_p=2\Omega$$

Step 2: Total Resistance

$$R_{\text{total}}=2+2=4\Omega$$

If connected to $12\text{ V}$:

$$I=\frac{12}{4}=3.0\text{ A}$$

Brightness and Power Reasoning

Lamp brightness depends on power dissipated.

$$P=IV$$

For identical lamps:

• greater current usually means brighter
• greater p.d. means greater power

Series Lamps

Same current, shared voltage.

Usually dimmer than a single lamp alone.

Parallel Lamps

Same supply voltage across each lamp.

Usually each lamp is brighter than in series arrangement.

Worked Example 2: Identical Lamps

Two identical lamps connected:

In Series Across Same Cell

Each receives smaller p.d. than supply.

In Parallel Across Same Cell

Each receives full supply voltage.

Hence lamps are brighter in parallel.

Meter Connection Rules

Ammeter

• connected in series
• very low resistance

Purpose: measure current without changing circuit significantly.

Voltmeter

• connected in parallel
• very high resistance

Purpose: measure p.d. across a component.

Common Errors

• ammeter in parallel may short circuit
• voltmeter in series may reduce current drastically

Potential Divider Overview

Two resistors in series divide the supply voltage in proportion to resistance.

For $R_1$ and $R_2$ across supply $\mathcal{E}$:

$$V_1=\mathcal{E}\frac{R_1}{R_1+R_2}$$ $$V_2=\mathcal{E}\frac{R_2}{R_1+R_2}$$

Applications:

• adjustable output voltage
• sensor circuits
• control knobs

See Potential Divider.

Thermistor / LDR Divider Behaviour

See also Thermistors and LDRs.

Thermistor

Resistance changes with temperature.

LDR

Resistance changes with light intensity.

When used in a divider, output voltage changes with environment.

Examples:

• thermostats
• automatic street lights
• alarms

Potentiometer Overview

A potentiometer uses a uniform wire carrying current to produce a known potential gradient.

Unknown emf is balanced against wire voltage.

At balance:

• galvanometer reads zero
• no current flows in test branch
• measurement is accurate

This is the null method.

See Potentiometer.

Internal Resistance in Full Circuits

Real cells have internal resistance $r$.

Terminal p.d.:

$$V=\mathcal{E}-Ir$$

Meaning:

• larger load current gives larger lost volts
• terminal p.d. falls when current increases

See Internal Resistance.

Circuit Fault Reasoning Overview

Common faults:

• open circuit
• short circuit
• broken lamp
• wrong meter placement

Typical clues:

• unexpected zero current
• zero p.d. across working resistor
• full supply across broken component

See Circuit Fault Finding.

Formula Summary

Ohm’s Law

$$V=IR$$

Power

$$P=IV$$ $$P=I^{2}R$$ $$P=\frac{V^2}{R}$$

Series Resistance

$$R_{\text{total}}=R_1+R_2+\cdots$$

Parallel Resistance

$$\frac{1}{R_{\text{total}}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdots$$

Internal Resistance

$$V=\mathcal{E}-Ir$$

Common Exam Pitfalls Overview

1. Wrong Series / Parallel Identification

Check junctions carefully.

2. Current vs p.d. Confusion

• current same in series
• p.d. same in parallel

3. Wrong Meter Placement

Ammeter in series, voltmeter in parallel.

4. Brightness Errors

Brightness depends on power, not current alone.

5. Divider Misuse

Use correct resistor in numerator.

6. Potentiometer Misconception

Zero galvanometer current does not mean zero current everywhere.

For a full checklist see DC Circuits Common Exam Traps.

Links

• Current Electricity Fundamentals
• Potential Divider
• Potentiometer
• Circuit Fault Finding
• DC Circuits Common Exam Traps
• Internal Resistance
• I-V Characteristics

Summary

Strong DC-circuit performance comes from mastering a few rules:

• current conservation at junctions
• potential changes around loops
• series and parallel reduction
• correct power reasoning
• correct meter use
• structured step-by-step solving

Use these consistently and most exam questions become manageable.