Physics · Electricity and magnetism · Grade 8

Circuits and Kirchhoff's rules

Why are the sockets at home connected in parallel, and how does current split at a junction?

The situation

In the college laboratory stands a mounting board: lamps, resistors, switches, galvanic cells, an ammeter and a voltmeter. An electrician wiring a home relies on the same rules: sockets are connected in parallel, and the circuit breaker watches the total current. What happens if we connect a voltmeter in series or an ammeter in parallel? Build the circuit yourself and measure the current and voltage across every element: how does current divide in a branched circuit?

Topic

Building electric circuits, Kirchhoff's rules. Flat wiring: ammeter, voltmeter and a branched circuit

Key ideas

Kirchhoff's rules
Shunt and series resistor
Wheatstone bridge

Formulas

\sum_{k} I_{k} = 0

Kirchhoff's 1st rule (junction): the sum of currents entering a junction equals the sum of currents leaving it — charge does not accumulate.

\sum_{k} ε_{k} = \sum_{k} I_{k} R_{k}

Kirchhoff's 2nd rule (closed loop): the sum of EMFs equals the sum of voltage drops (the internal resistance of the sources is also included).

R = R_{1} + R_{2}, \frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}}, I = \frac{ε}{R + r}

Series and parallel connection; Ohm's law for a complete circuit.

R_{s h} = \frac{R_{g} I_{g}}{I - I_{g}}, R_{q} = \frac{U}{I_{g}} - R_{g}

Turning a galvanometer into an ammeter measuring up to $I$ (shunt) and a voltmeter measuring up to $U$ (series resistor).

R_{x} = R_{3} \frac{R_{2}}{R_{1}}

The Wheatstone bridge in balance ( $I_{G} = 0$ ).

Theory

Building the circuit and measuring

An ammeter is connected in series with the element being measured — the current must flow through it, so its resistance is very small. A voltmeter is connected in parallel across the element, and its resistance is very large. In a series connection the current is the same everywhere and the voltages add: $U = U_{1} + U_{2}$ . In a parallel connection the voltage is the same and the currents add: $I = I_{1} + I_{2}$ . In a mixed connection we split the circuit into parts: first find the resistance of the parallel part, then add in series. The blue dots on the board move in the conventional current direction (from the “+” terminal of the source to “−”), at a speed proportional to the current.

Kirchhoff's rules

A circuit with two sources cannot be simplified with the series/parallel rules. Rule 1 follows from conservation of charge: charge does not accumulate at a junction, $\sum I = 0$ . Rule 2 follows from the single-valuedness of potential: going round a closed loop back to the starting point, the sum of potential changes is zero, i.e. $\sum ε = \sum I R$ . The simulator solves the circuit by nodal analysis (MNA): rule 1 is written for each node, and the unknown potentials are found from the system of equations. In “Select” mode click a terminal — the currents at the junction appear; the “Next loop” button counts the voltages around the loop.

A non-uniform circuit and connecting sources

A source also has resistance inside it: the voltage across its terminals is $U = ε - I r$ . Ohm's law for a section with an EMF: $I R_{12} = φ_{1} - φ_{2} + ε_{12}$ ; for a closed circuit $I = ε / (R + r)$ . If identical cells are connected in series, the EMFs and internal resistances add ( $n ε$ , $n r$ ); if they are connected in parallel, the EMF is unchanged and the internal resistance falls to $r / n$ — the battery can supply a large current, but the voltage does not increase. Compare them in the “Sources” circuit. At the micro level Ohm's and Joule–Lenz laws are written as $j = σ E$ , $w = σ E^{2}$ — this is in the “Electric current in metals” experiment.

Designing measuring instruments

A galvanometer deflects fully at a very small current ( $I_{g}$ = 100 µA). To turn it into an ammeter for large currents, a shunt is connected in parallel: most of the current goes through the shunt, $R_{s h} = R_{g} I_{g} / (I - I_{g})$ . To make a voltmeter, a series resistor is connected in series: $R_{q} = U / I_{g} - R_{g}$ . Real instruments affect the circuit: the $R_{A}$ of an ammeter reduces the current, and the finite $R_{V}$ of a voltmeter itself draws current. In the “Real instruments” circuit compare the two connections for the $R = U / I$ method: for a small $R_{x}$ it is more accurate to connect the voltmeter across $R_{x}$ alone, and for a large $R_{x}$ together with the ammeter.

Wheatstone bridge

In the bridge two voltage dividers are connected in parallel, and a galvanometer joins their midpoints. When $R_{1} / R_{2} = R_{3} / R_{x}$ , these points are at the same potential and no current flows through the galvanometer. This is the null method: the error of the instrument's scale does not affect the result, and the accuracy depends on that of $R_{1}$ , $R_{2}$ , $R_{3}$ . Choose the ratio $R_{2} / R_{1}$ (0.1; 1; 10) so that more digits of the resistance box $R_{3}$ are in use. The bridge circuit is used in electronic scales with strain gauges and in resistance thermometers.

The home wiring network and engineering

Sockets and lamps in a house are connected in parallel: each gets 220 V and does not depend on the others, and the total current is $I = \sum P_{k} / U$ . A 16 A circuit breaker withstands about $220 \cdot 16 \approx 3.5$ kW. In an overload the bimetallic strip of the breaker heats up and trips with a slight delay, while in a short circuit ( $I \geq 5 I_{n}$ ) an electromagnet breaks the circuit instantly. Model: the network is calculated with rms values as if it were direct current, appliances are simple resistors ( $R = U^{2} / P$ ), the lamp resistance is constant at working temperature, wire resistance is zero, and the heating of the bimetal is simplified ( $τ$ = 20 s). More in the “Electric circuit, Ohm's law” experiment.

Did you know?

Gustav Kirchhoff published his circuit rules in 1845, while still a student at the University of Königsberg.
The bridge circuit was invented in 1833 by Samuel Christie; Charles Wheatstone improved it in 1843 and made it widely known, so the bridge bears his name.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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