D.C. Circuits: Kirchhoff's Laws and Potential Dividers

This guide covers Topic 10, D.C. circuits, in full — subtopics 10.1 Practical circuits, 10.2 Kirchhoff’s laws and 10.3 Potential dividers — from Cambridge International AS & A Level Physics 9702, 2025–2027 series. This is AS Level content.

Before studying this

This resource assumes current, potential difference, power and resistance from Electricity: Current, Potential Difference and Resistance.

Syllabus coverage

CAMBRIDGE INTERNATIONAL AS & A LEVEL PHYSICS 9702 — AS Level, Topic 10

10.1 Practical circuits — recalling and using standard circuit symbols; drawing and interpreting circuit diagrams; defining e.m.f. as energy transferred per unit charge driving charge around a complete circuit; distinguishing e.m.f. from potential difference; understanding the effect of internal resistance on terminal potential difference.

10.2 Kirchhoff’s laws — recalling Kirchhoff’s first law as a consequence of charge conservation and the second as a consequence of energy conservation; deriving and using formulas for combined resistance in series and parallel; using Kirchhoff’s laws to solve simple circuit problems.

10.3 Potential dividers — the principle of a potential divider circuit; recalling and using the potentiometer principle for comparing potential differences; understanding a galvanometer’s use in null methods; explaining the use of thermistors and LDRs in potential dividers to give a temperature- or light-dependent output.

Circuit symbols

Subtopic 10.1 asks you to recall and use the circuit symbols listed in the 9702 syllabus, and to draw and interpret circuit diagrams that contain them. The syllabus’s own table (“Circuit symbols”, in the additional information at the back of the 2025–2027 syllabus) lists the 27 symbols below. It calls the table a guide to the symbols that may be used in examination papers. The descriptions are in words; check each one against the drawn table in the official syllabus, and practise drawing them.

Component	How the symbol is drawn
Cell	Two parallel lines across the wire, one long line and one short line (drawn the same thickness). The long line is the positive terminal.
Battery of cells	Two or more cell symbols side by side, all facing the same way; or two cells joined by a dashed line to show that there are more cells in between.
Power supply	Two small open circles (the terminals), each at the end of a short lead, with a gap between them.
a.c. power supply	The power-supply symbol with a wavy line (∼) in the gap between the two terminals.
Junction of conductors	A solid dot where one wire meets another and is joined to it.
Lamp	A circle with a cross (×) inside it, the ends of the cross touching the circle.
Fixed resistor	An open rectangle in line with the wire.
Variable resistor	The resistor rectangle with a diagonal arrow drawn through it, the arrowhead at the upper end.
Thermistor	The resistor rectangle with a diagonal line through it (no arrowhead) that bends into a short horizontal “tail” at its lower end.
Light-dependent resistor (LDR)	The resistor rectangle with two parallel arrows pointing towards it (light falling on it).
Heater	A rectangle divided into several equal sections by short vertical lines.
Potentiometer	The resistor rectangle with an arrow pointing onto its long side from a third lead (the sliding contact).
Diode	A triangle pointing along the wire, with a short line across the wire at the triangle’s tip. The triangle points in the direction of conventional current that the diode allows.
Light-emitting diode (LED)	The diode symbol with two parallel arrows pointing away from it (light given out).
Switch	A gap in the wire, with a short line hinged at one contact and angled away from the other (switch open).
Earth	A lead ending in three horizontal lines, one below another, each shorter than the one above.
Electric bell	A semicircle with its curved side upwards on a flat base, with two leads going down from the base.
Buzzer	A semicircle the other way up (flat side at the top, curved side below), with two leads going down from it.
Microphone	A circle with a straight line touching one side of it, and two leads leaving from the other side.
Loudspeaker	A small rectangle with a flared shape (like a cone) on one side, and two leads entering the rectangle from the other side.
Motor	A circle containing the letter M, with a short bracket under the M.
Generator	A square containing the letter G.
Ammeter	A circle containing the letter A.
Voltmeter	A circle containing the letter V.
Galvanometer	A circle containing a vertical arrow pointing upwards (a pointer).
Oscilloscope	A circle containing a sawtooth-shaped trace.
Capacitor	Two parallel lines of equal length across the wire, with a gap between them.

Pairs that are easy to confuse:

• Cell and capacitor — both are two parallel lines. A cell’s lines are of unequal length; a capacitor’s are equal.
• Variable resistor and potentiometer — an arrow through the rectangle means a variable resistor (two connections); an arrow pointing onto the rectangle from a third lead means a potentiometer (a sliding contact that taps off part of the p.d., as in the potential dividers below).
• Variable resistor and thermistor — the thermistor’s diagonal line has a horizontal tail and no arrowhead. Its resistance changes with temperature, not by hand.
• LDR and LED — the arrows point in for a light-dependent resistor (light received) and out for a light-emitting diode (light given out).
• Motor and generator — M in a circle, but G in a square.
• Power supply and a.c. power supply — the only difference is the wavy line between the terminals.

Drawing and reading circuit diagrams. Draw wires as straight lines with square corners, and put a junction dot wherever wires are joined. An ammeter goes in series with the component whose current it measures; a voltmeter goes in parallel across the component whose p.d. it measures (see Meters below). When you interpret a diagram, trace each loop from the supply and back, and identify every junction before applying Kirchhoff’s laws.

E.m.f., internal resistance and terminal p.d.

Electromotive force (e.m.f.) is the energy transferred per unit charge by a source in driving charge around a complete circuit — it is defined for the source, whether or not current is flowing, and is distinct from potential difference, which is the energy transferred per unit charge across any component.

Real sources have internal resistance r. While a source is discharging — driving current I through an external circuit — some energy is transferred within the source itself, so the terminal potential difference (what’s actually available to the external circuit) is less than the e.m.f. While a source is instead being charged (current driven into its positive terminal, for example by another, higher-voltage source), the opposite holds: the applied terminal voltage exceeds the e.m.f. by Ir. The discharging case is the one usually met first:

ε = I(R + r) = V_terminal + Ir

This is why a battery’s measured terminal voltage drops under load, especially noticeable when internal resistance is significant relative to the external circuit’s resistance.

Worked example. A cell of e.m.f. 1.50 V and internal resistance 0.80 Ω is connected to a 4.20 Ω resistor. Current I = ε / (R + r) = 1.50 / (4.20 + 0.80) = 1.50 / 5.00 = 0.30 A. Terminal p.d. = ε − Ir = 1.50 − (0.30 × 0.80) = 1.50 − 0.24 = 1.26 V. If the external resistance is reduced, current increases, so more energy is transferred inside the cell (Ir increases) and the terminal p.d. falls further below the e.m.f.

Reading e.m.f. and internal resistance from a V-I graph. Since V_terminal = ε − Ir, plotting terminal p.d. against current gives a straight line with a negative gradient equal to −r and a y-intercept equal to ε. A graph with gradient −0.45 V A⁻¹ and intercept 1.62 V therefore gives ε = 1.62 V and r = 0.45 Ω.

Kirchhoff’s laws

Kirchhoff’s first law: the sum of currents entering a junction equals the sum of currents leaving it — a direct consequence of conservation of charge.

Kirchhoff’s second law: around any closed loop in a circuit, the sum of e.m.f.s equals the sum of potential differences — a direct consequence of conservation of energy.

Combined resistance. Using Kirchhoff’s laws:

• Series: R_total = R₁ + R₂ + R₃ + …
• Parallel: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + …

Worked example. Two resistors, 6.0 Ω and 3.0 Ω, are connected in parallel:

1/R = 1/6.0 + 1/3.0 = 1/6.0 + 2/6.0 = 3/6.0
R = 2.0 Ω

Potential dividers

A potential divider splits a supply voltage across two (or more) series components in proportion to their resistance, producing a chosen fraction of the supply voltage at the junction between them. This is the basis of the potentiometer, which can be used to compare potential differences, and null methods, where a galvanometer detects zero current (and hence a balanced condition) rather than measuring a current directly — a more precise technique than a direct voltmeter reading in many cases.

Replacing one fixed resistor in a potential divider with a thermistor or LDR produces an output voltage that varies with temperature or light intensity respectively — a light- or temperature-sensing circuit, since the varying component’s changing resistance changes the voltage division ratio.

Meters

An ideal ammeter has zero resistance and is connected in series, so it does not reduce the current it is measuring. An ideal voltmeter has infinite resistance and is connected in parallel, so it draws no current away from the component it is measuring. A real voltmeter draws a small current, since its resistance is large but not infinite, which slightly reduces the potential difference it is trying to measure — a systematic error worth naming explicitly when a question asks why a measured reading differs from a calculated one.

Common mistakes

• Treating e.m.f. and terminal potential difference as the same thing. They are only equal when internal resistance is negligible or no current flows (open circuit).
• Adding resistances in parallel directly (e.g. writing R = R₁ + R₂ for a parallel combination) — parallel resistors combine via reciprocals, always giving a combined resistance smaller than the smallest individual resistor.
• Forgetting that Kirchhoff’s second law includes e.m.f.s as well as potential differences around a loop — omitting a battery’s e.m.f. from the loop equation is a frequent source of error.
• Not tracing through which component’s resistance change increases or decreases the potential-divider output — work through the ratio explicitly rather than guessing the direction.

Quick revision checklist

• The 27 circuit symbols in the syllabus table, and the pairs that are easy to confuse
• E.m.f. vs. potential difference, and the effect of internal resistance on terminal p.d.
• Kirchhoff’s first and second laws, and their conservation-law basis
• Combined resistance formulas for series and parallel
• The potential divider principle, and thermistor/LDR sensing circuits

Related resources

• Electricity: Current, Potential Difference and Resistance — the previous AS topic
• Particle Physics: Atoms, Nuclei and Fundamental Particles — the final AS topic
• Cambridge AS & A Level Physics hub

Written against Cambridge International AS & A Level Physics 9702, 2025–2027 series. Always check the current syllabus for your examination year.