Physics · Electricity and magnetism · Grade 7

Power sources, EMF, internal resistance

Why can't a car battery be short-circuited, while a lemon cannot light a lamp?

The situation

On a cold winter morning, when the driver turns the key, the starter barely turns and the headlights dim. The accumulator is marked “12 V”, but when the starter draws hundreds of amperes from it, the voltage across its terminals drops sharply. At a school club, students made a battery by sticking zinc and copper plates into a lemon: the voltmeter read 0.9 V, but a small lamp did not light. What is “inside” a source of current — how do the emf and internal resistance determine the current and power in a circuit?

Topic

Current sources, EMF and internal resistance. Lemon battery, car accumulator and short circuit

Key ideas

Ohm's law for a complete circuit
EMF and internal resistance
Efficiency of a source

Formulas

ε = \frac{A_{ext}}{q}

Emf is the work done by external (non-electrostatic) forces inside the source in moving 1 C of positive charge.

I = \frac{ε}{R + r}, I_{qt} = \frac{ε}{r}

Ohm's law for a complete circuit and the short-circuit current ( $R = 0$ ).

U = ε - I r, I R_{12} = φ_{1} - φ_{2} + ε_{12}

Terminal voltage (when charging, the current is reversed: $U = ε + I r$ ) and Ohm's law for a non-uniform section of a circuit.

P = \frac{ε^{2} R}{{(R + r)}^{2}}, η = \frac{R}{R + r}

Useful power is greatest at $R = r$ ( $ε^{2} / 4 r$ ), and the efficiency is then 50 %.

ε_{b} = n ε, r_{b} = \frac{n r}{m}

A battery with $n$ cells in series in each of $m$ groups connected in parallel.

Theory

Sources of current and external forces

An electrostatic field does zero work when it carries a charge round a closed path — so electrostatic forces alone cannot sustain a current in a closed circuit. Inside the source, non-electrostatic external forces do the work: chemical (galvanic cell, accumulator), light (solar cell), magnetic (generator — the “Electromagnetic induction” experiment), thermal (thermocouple). They carry positive charge against the field — from the negative terminal to the positive one, just as a pump lifts water uphill. The emf $ε = A_{ext} / q$ is the energy given to each coulomb of charge; its unit is the volt. In “Where the energy comes from” mode, look at the two forces on an ion and the pump analogy.

Ohm's law for a complete circuit

The current is determined by the resistance of the whole circuit: $I = ε / (R + r)$ . In the external (source-free) part $U = I R$ , while at the terminals of the source $U = ε - I r$ : the more current, the more the voltage drops — when the starter draws hundreds of amperes, the headlights dim. With the switch open ( $I = 0$ ) the voltmeter reads the emf. As $R \to 0$ the short-circuit current is $I_{sc} = ε / r$ : ≈ 7.5 A for an AA battery, over a thousand amperes for an accumulator (the wires heat up and melt!), and only ≈ 1 mA for a lemon. The “Potential diagram” shows the potential rising by $ε$ inside the source, dropping by $I r$ across the internal resistance, and falling to zero along the rheostat in the external circuit. When an accumulator is charged, the current flows backwards through the source and $U = ε + I r$ — a special case of Ohm's law for a non-uniform section, $I R_{12} = φ_{1} - φ_{2} + ε_{12}$ .

Power and efficiency

The power of the external forces $ε I$ splits into two parts: the useful power in the load $I^{2} R$ and the heat released inside the source $I^{2} r$ (the Joule–Lenz law $Q = I^{2} R t$ ). The graph of $P (R)$ has a maximum at $R = r$ : $P_{\max} = ε^{2} / 4 r$ , but the efficiency $η = R / (R + r)$ is then only 50 %. For a high efficiency we need $R ≫ r$ — which is why sources are built with a small internal resistance. For a solar panel the $U$ – $I$ relation is not a straight line: measure it and find the point of maximum power.

Connecting cells and Kirchhoff's rules

When cells are connected in series, the emfs and internal resistances add: $n ε$ , $n r$ . When identical cells are connected in parallel, the emf does not change and the internal resistance falls $m$ -fold. In a mixed arrangement $ε_{b} = n ε$ , $r_{b} = n r / m$ ; for a given number of cells the current is greatest when $n r / m \approx R$ . At a junction the sum of the currents entering equals the sum leaving (Kirchhoff's 1st rule: each group carries a current $I / m$ ), and around a closed loop $\sum ε = \sum I R$ (2nd rule). Branched circuits are covered in the “Building electric circuits” experiment.

Chemical and alternative sources

In a galvanic cell zinc goes into solution as ions: $Zn \to {Zn}^{2 +} + 2 e^{-}$ , and the electrons stay on the zinc and make it negative. At the copper, hydrogen ions take electrons and become gas bubbles: $2 H^{+} + 2 e^{-} \to H_{2}$ . The same process happens in a lemon — the copper does not dissolve, and the lemon juice acts as the electrolyte. Inside the solution the current is carried by ions (details in the “Electrolysis” experiment). In an accumulator the reaction is reversible: during charging an external source drives it backwards; a 12 V, 60 A·h accumulator stores ≈ $12 \cdot 60 \cdot 3600 \approx 2.6$ MJ of energy. Alternative sources: the solar cell (the photoeffect at a p–n junction — the “Semiconductors” experiment), wind and water generators, thermocouples.

Measuring emf and internal resistance; the model

1) With an open circuit the voltmeter reads the emf, provided its resistance $R_{V} ≫ r$ . 2) ** $U$ – $I$ graph**: record points at different $R$ ; the straight line $U = ε - r I$ meets the $U$ axis at $ε$ , and its gradient is $r$ . 3) Compensation method: in a uniform wire the voltage is proportional to the length; when the galvanometer reads zero no current flows through the cell being tested, so its $r$ has no effect: $ε_{x} = ε_{n} L_{x} / L_{n}$ (Weston standard $ε_{n}$ = 1.0186 V). Model: the ammeter and voltmeter are ideal (except in the comparison in compensation mode), $r$ is constant, the emf of the accumulator depends linearly on its state of charge and the process is sped up 600 times, the $r$ of the lemon and of the Volta cell is approximate, and the lamp resistance is constant.

Did you know?

The first chemical source of current — the “voltaic pile”, with cardboard soaked in salt water placed between zinc and copper discs — was created in 1800 by the Italian scientist Alessandro Volta; the unit of voltage, the volt, is named in his honour.
A car accumulator consists of 6 lead-acid cells connected in series: each fully charged cell gives about 2.1 V, in total ≈ 12.6 V.

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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Electrification, Coulomb's law — Grade 7
Electric circuit, Ohm's law — Grade 7
Conductor, dielectric, lightning — Grade 7
Work and power of an electric current — Grade 8
Electric motor — Grade 8
Aurora and the Lorentz force — Grade 8
Electric field and potential — Grade 8
Capacitor and capacitance — Grade 8
Circuits and Kirchhoff's rules — Grade 8
Electric current in metals — Grade 8