DC Circuits

Topic hub: This is the main overview page for Topic 14. Use it as the starting point, then follow the branch notes for deeper treatment of potential dividers, potentiometers, circuit fault finding, and common exam traps.

Syllabus focus

The repository calls this Topic 14, while the 2026 H2 Physics 9749 syllabus lists it as Topic 15, D.C. Circuits. The examinable core is circuit symbols, series and parallel networks with one source, potential dividers, thermistor/LDR divider applications, and the potentiometer principle. Clearly labelled application or enrichment material supports these outcomes without extending the required scope.

Overview

This topic extends Current Electricity Fundamentals into complete direct-current circuits with sources, resistors, lamps, meters, dividers, and potentiometers.

The key skill is not just substitution into formulas. Students need to read the circuit first, then decide which local rule applies.

The most reliable sequence is:

$$\text{identify nodes and paths}⟶\text{decide the topology}⟶\text{apply local current and p.d. rules}⟶\text{check the result physically}.$$

Main skills:

• read circuit symbols and component layouts
• analyse series, parallel, and mixed networks
• track current conservation at junctions
• track potential balance around loops and between nodes
• compare lamp brightness using power
• use divider logic for variable output and sensor circuits
• apply the potentiometer null method
• diagnose open-circuit, short-circuit, and meter-connection faults

Circuit Symbols and Diagram Reading

Symbol literacy matters because later circuit questions assume the diagram has already been interpreted correctly.

The core symbols in this topic include:

• cell or battery
• resistor
• lamp
• ammeter
• voltmeter
• variable resistor
• thermistor
• LDR
• switch
• galvanometer
• earth
• potential divider

Why this matters

The circuit shape controls which quantities are shared:

• series components share current
• parallel branches connected across the same two nodes share the same p.d.
• ideal wires share the same potential

Figure: Standard symbols encode both component identity and circuit connectivity. A filled node means conducting wires join; a crossing without a node means the paths are not electrically connected. The long line of a cell symbol is its positive terminal.

Core Ideas

DC-circuit analysis rests on a small set of ideas that recur throughout the topic:

• current is conserved at junctions
• potential difference is energy transferred per unit charge
• resistance controls how current and p.d. are distributed
• power determines heating and lamp brightness
• balanced potentials imply zero current through the galvanometer branch, not zero current in the whole circuit

Circuit Vocabulary: Node, Branch, Path, and Loop

• A node is a set of points joined by ideal wire and therefore at the same potential.
• A branch is a conducting route between two nodes; it may contain one or more components.
• A path is any chosen route through connected components.
• A loop is a closed path returning to its starting node.

Two components are in series only when the same current must pass through both and there is no branch between them. Two components are in parallel only when their two terminals connect to the same pair of nodes. Their visual orientation on the page is irrelevant.

Potential, Potential Difference, and Reference Nodes

Potential and potential difference should not be merged into one vague idea.

• Electric potential is a value assigned to a point in a circuit relative to a chosen reference point.
• Potential difference is the difference in electric potential between two points.
• In circuit energy terms, the potential difference across a passive component is the electrical energy transferred to other forms per unit charge as charge passes through that component.
• All points connected by an ideal wire are at the same potential, so there is no p.d. between them.
• A chosen reference node may be assigned $0\mathrm{V}$, often called ground.
• What matters physically is not the absolute value of potential, but potential difference. Therefore, we are free to choose a convenient reference node; assigning it $0\mathrm{V}$ is the usual choice.

This is why potential-divider questions often ask for the potential of a point relative to the reference node, while energy-transfer questions often ask for the p.d. across a component.

Core Quantities

Current

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

Potential Difference

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

Notation Note: $V$ versus $\Delta V$

Strictly, electric potential is a node value, while potential difference is the difference between two node potentials:

We use the signed convention

$$V_{AB}=V_A-V_B.$$

Thus $V_{AB}$ is positive when point A is at higher potential than point B. Some questions use “p.d. across a component” as a positive magnitude, so state a polarity or reference direction whenever a sign matters.

However, in circuit physics, it is conventional to use $V$ to denote the potential difference across a component or between two stated points:

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

This is a shorthand convention. Whenever $V$ appears in circuit formulae such as $V=IR$ or $P=IV$, it usually means the p.d. across the component, not the absolute potential at one node.

A frequent source of confusion is that the same symbol $V$ is also sometimes used to denote the electric potential at a point. The meaning must therefore be read from context.

Resistance

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

The operating resistance is defined by $R=V/I$, so $V=IR$ may be used at a stated operating point. Ohm’s law makes the stronger claim that $V\propto I$ and $R$ remains constant when physical conditions, especially temperature, are constant.

Power

$$P=IV$$

Also:

$$P=I^{2}R$$

Since $I=V/R$

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

Exam Relevance

DC-circuit questions are usually won or lost on structure, not algebra. The key is to identify the network correctly, assign current and p.d. locally, and then apply the appropriate series, parallel, junction, or loop rule without treating formulas as global shortcuts.

Series and Parallel Circuits

Series

For components in series:

$$R_{\text{eff}}=R_1+R_2+\cdots +R_N$$

The same current flows through each component, while the total p.d. is shared across the chain. Since $V_i=I{R}_i$ and $V=\sum V_i$,

$$V=I(R_1+R_2+\cdots )=I{R}_{\mathrm{eff}},$$

which gives the series formula. For positive resistances, $R_{\mathrm{eff}}$ must exceed the largest individual resistance.

Parallel

For components in parallel:

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

The same p.d. is across each branch, while the current splits between branches. Since $I_i=V/R_i$ and $I=\sum I_i$,

$$I=V\left(\frac{1}{R_1}+\frac{1}{R_2}+\cdots \right)=\frac{V}{R_{\mathrm{eff}}}.$$

For positive resistances, a parallel equivalent resistance must be smaller than the smallest branch resistance.

First numerical checks

Two resistors of $2.0\Omega$ and $4.0\Omega$ in series have

$$R_{\mathrm{eff}}=2.0+4.0=6.0\Omega .$$

Across a $12\mathrm{V}$ supply, the common current is $I=12/6.0=2.0\mathrm{A}$, and the component p.d.s are $4.0\mathrm{V}$ and $8.0\mathrm{V}$. They add to the supply p.d.

The same two resistors in parallel have

$$R_{\mathrm{eff}}={\left(\frac{1}{2.0}+\frac{1}{4.0}\right)}^{-1}=1.33\Omega .$$

Across $12\mathrm{V}$, their branch currents are $6.0\mathrm{A}$ and $3.0\mathrm{A}$. The total current is $9.0\mathrm{A}$, the sum of the branch currents.

Figure: Series is a one-path condition, so every component carries the same current. Parallel is a same-two-nodes condition, so every branch has the same p.d. The resistance checks shown catch many topology or algebra errors.

Junctions and Loops

At a junction:

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

This is conservation of charge.

Around a closed loop, the potential rises and drops must balance because a charge returning to its starting node returns to the same potential. This energy-per-charge statement is useful for one-source circuits even though general multi-source Kirchhoff equation systems are beyond this topic’s required scope.

Mixed Networks

Most exam questions are not pure series or pure parallel. The key is to reduce them step by step.

Method

1. identify an obvious series group
2. identify an obvious parallel group
3. replace one group with its equivalent resistance
4. find the total current from the source
5. work backwards to recover branch currents or p.d.s

Worked Example

Figure: The $3.0\Omega$ and $6.0\Omega$ resistors share the same two nodes, so they are parallel and combine to $2.0\Omega$. That equivalent is in series with the separate $2.0\Omega$ resistor, giving $R_{\mathrm{eff}}=4.0\Omega$.

Worked Mixed-Network Example

With $12\mathrm{V}$ across this network,

$$I_{\mathrm{total}}=\frac{12}{4.0}=3.0\mathrm{A}.$$

The series $2.0\Omega$ resistor has p.d.

$$V=IR=(3.0)(2.0)=6.0\mathrm{V}.$$

The parallel pair therefore also has $6.0\mathrm{V}$ across it, so

$$I_{3\Omega }=\frac{6.0}{3.0}=2.0\mathrm{A},I_{6\Omega }=\frac{6.0}{6.0}=1.0\mathrm{A}.$$

The branch currents add to $3.0\mathrm{A}$, checking charge conservation.

Brightness, Power, and Meter Rules

Figure: Lamp brightness is compared using operating electrical power, not by saying current “chooses” a path. An ideal ammeter has negligible resistance and is connected in series; an ideal voltmeter has very large resistance and is connected in parallel across the component or terminals being measured.

For otherwise identical lamps under comparable conditions, greater electrical power usually means greater brightness. A filament lamp is non-ohmic, so do not assume its resistance stays constant unless the question gives or implies that model. Ammeters and voltmeters must also be connected without materially changing the circuit being measured.

Useful reminders:

• a brighter lamp is usually dissipating more power
• if the current changes, the lamp power usually changes too
• if the p.d. across a lamp changes, its power changes
• in a series fault, an open circuit can make the whole loop go dark

Meter rules:

• ammeter in series
• voltmeter in parallel
• ideal ammeter has zero resistance; a real ammeter is designed to have very low resistance
• ideal voltmeter has infinite resistance; a real voltmeter is designed to have very high resistance

These rules matter because a wrongly connected meter can change the circuit, not just measure it.

Fault Finding

Fault-finding questions use the same laws as normal network questions. The difference is that one component or connection is wrong.

Open circuit

A break in the conducting path means:

• no current in that loop
• lamps in the same series loop go off
• a voltmeter across an intact component carrying no current may read $0\text{ V}$ if there is no p.d. across it
• a voltmeter across the broken component may read the full supply p.d., depending on the rest of the circuit

Short circuit or bypass

A very low resistance path around a component means:

• the low-resistance bypass makes the p.d. across the parallel component nearly zero
• the bypassed component gets little or no p.d.
• lamp brightness falls sharply

Figure: An open fault removes a conducting path and can support a large p.d. across the gap. An ideal zero-resistance short forces the two bypass nodes to the same potential; a practical low-resistance bypass greatly reduces, but may not make exactly zero, the p.d. across the component.

Potential Dividers and Reference Potential

A potential divider is a series-resistor circuit that splits a supply p.d. in proportion to the resistances.

For two series resistors with actual p.d. $V_{\mathrm{in}}$ across the unloaded divider:

$$V_{AB}=\frac{R_1}{R_1+R_2}{V}_{\mathrm{in}}$$ $$V_{BC}=\frac{R_2}{R_1+R_2}{V}_{\mathrm{in}}$$

and:

$$\frac{V_{AB}}{V_{BC}}=\frac{R_1}{R_2}$$ *Figure: The numerator is the resistance between the two output terminals. The simple rule assumes the output is unloaded. Set $V_{\mathrm{in}}=\mathcal E$ only for an ideal source or when source internal resistance is negligible.*

Reference potential

• a node can be assigned a potential such as $0\text{ V}$ by grounding it
• p.d. is the difference between the potentials of two points
• all points on an ideal wire have the same potential
• a node can be positive or negative relative to the chosen reference

Potential-Distance Graph

Key interpretation:

• potential stays constant along ideal wires
• potential falls across passive components in the direction of conventional current when the component is dissipating electrical energy
• potential increases across a source such as a battery
• grounding one point sets the zero reference

Figure: The labelled circuit and graph follow the same direction around one loop. Potential is flat along ideal wires, falls through passive resistors in the direction of current, and rises from the negative to positive terminal of the ideal source. The chosen zero merely fixes the reference level.

Variable-Resistor and Sensor Dividers

The divider output changes when one resistance changes.

This is the basis of variable-output circuits and sensor circuits.

See also Thermistors and LDRs.

NTC thermistor

For the negative-temperature-coefficient (NTC) thermistor used in this syllabus context, resistance decreases as temperature increases.

LDR

LDR resistance decreases as light intensity increases.

The exact p.d. change depends on where the sensor is placed in the divider:

• if the sensor is the output resistor, decreasing resistance may lower the output p.d.
• if the sensor is the other resistor, decreasing resistance may raise the output p.d.

That is why the circuit layout matters as much as the device identity.

Figure: With output measured from the centre node to $0\mathrm{V}$, decreasing the lower resistance decreases $V_{\mathrm{out}}$, while decreasing the upper resistance increases $V_{\mathrm{out}}$. Apply this placement rule after identifying how the NTC or LDR resistance changes.

Figure: A three-terminal sliding-contact divider selects a continuous fraction of the input p.d. With the lower end chosen as $0\mathrm{V}$, moving the slider from the lower to the upper end changes the unloaded output from $0$ to $V_{\mathrm{in}}$.

Potentiometers

The potentiometer uses the null method.

A uniform wire carrying a steady current has a uniform potential gradient:

$$V=kl$$

where $k$ is the potential gradient and $l$ is the length measured along the potentiometer wire. At the null point, $l$ is the balance length.

At balance:

• the galvanometer reads zero
• no current flows through the galvanometer and test cell
• the potentials at the two connected points are equal

This balances the unknown against a known wire drop instead of forcing current through the unknown cell.

Figure: The test source is connected so its p.d. opposes the potentiometer-wire drop. At balance the galvanometer terminals have equal potential, so $I_G=0$ while the driver circuit still carries current. A balance can occur only if the test p.d. does not exceed the maximum drop along the available wire.

Figure: The wire relation $V=kl$ is linear. A smaller potential gradient gives a longer balance length for the same test p.d. and therefore greater length sensitivity, but it also reduces the maximum p.d. that can be balanced on a wire of fixed length.

Figure: With the driver current unchanged, the same potential gradient $k$ applies to both null readings. Therefore $V_1=k{l}_1$, $V_2=k{l}_2$, and $V_1/V_2=l_1/l_2$; the galvanometer/test branch carries zero current at each balance.

Application Link: Internal Resistance in Full Circuits

Real cells have internal resistance $r$. When a source supplies current,

Terminal p.d. is:

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

So:

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

The external network receives the terminal p.d., not automatically the emf. For an external equivalent resistance $R$, use

$$I=\frac{\mathcal{E}}{R+r},V_{\mathrm{terminal}}=IR.$$

See Internal Resistance.

Quick Revision Summary

• series: current same, p.d. shared, resistances add
• parallel: p.d. same, current splits, reciprocals add
• junction rule: current in equals current out
• loop rule: rises and drops balance
• brightness: compare power, not current alone
• divider: voltage share follows resistance share
• potentiometer: null method, balance length, potential gradient
• faults: open circuit breaks the conducting path; short circuit bypasses the component

Common Exam Traps

See DC Circuits Common Exam Traps for the compact checklist.

The most common mistakes are:

• calling two components parallel when they do not share both end nodes
• swapping the series and parallel rules for current and p.d.
• using the total supply p.d. for a single component in a series chain
• treating a potentiometer null as if the whole circuit carries no current
• forgetting that a voltmeter in series can disrupt the circuit

Links

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