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.

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

• 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.