How a Potentiometer Measures a Cell's EMF Without Drawing Any Current

Many real circuits contain several cells and loops that no series or parallel shortcut can reduce. Two rules based on conservation of charge and energy solve all of them, and the same ideas power precise instruments for measuring resistance and emf.

This lesson covers Kirchhoff's laws, the Wheatstone and metre bridges, and the potentiometer.

**Kirchhoff's junction rule says the currents entering a junction equal the currents leaving it, and the loop rule says the sum of emfs around any closed loop equals the sum of the potential drops, $\sum \epsilon =\sum IR$.

What each rule conserves:

- Junction rule — charge, since none piles up at a junction
- Loop rule — energy, since a charge returning to its starting point has the same potential

The method. Mark a current in every branch with a guessed direction, use the junction rule to cut the unknowns, then write one loop equation per independent loop. A negative answer simply means the real current flows the other way.

Worked example.** A 12 V cell in series with $2.0\Omega$ and a 6.0 V cell in series with $3.0\Omega$ both feed a shared $6.0\Omega$ resistor, with currents $I_1$, $I_2$ and $I_3=I_1+I_2$.

$$12=2I_1+6(I_1+I_2)6=3I_2+6(I_1+I_2)$$

These simplify to $8I_1+6I_2=12$ and $6I_1+9I_2=6$, giving

$$I_1=2.0\text{ A}{I}_2=-0.67\text{ A}{I}_3=1.33\text{ A}$$

The negative $I_2$ shows the 12 V cell is actually charging the 6.0 V cell.

An everyday example. Jump-starting a car from another car's battery creates exactly this kind of two-cell circuit, with one battery charging the other.

The substance. Sign conventions matter more than guessed directions — crossing a resistor along the current is a drop of IR, and crossing a cell from negative to positive is a gain of $\epsilon$.

**A Wheatstone bridge is balanced, with no current through the galvanometer, when $\frac{P}{Q}=\frac{R}{S}$, and a metre bridge uses a uniform 1 m wire as the ratio arms so that $\frac{X}{R}=\frac{l}{100-l}$, with l in cm.

Why balance works. At balance, the two ends of the galvanometer are at the same potential, so the ratio of resistances on one side must match the other, whatever the cell's emf.

Worked example 1 — Wheatstone bridge.** With $P=10\Omega$, $Q=20\Omega$ and $R=15\Omega$:

$$S=\frac{QR}{P}=\frac{20\times 15}{10}=30\Omega$$

Worked example 2 — metre bridge. The unknown X sits in the left gap and a $6.0\Omega$ resistor in the right gap. The null point is at 40.0 cm:

$$X=R\frac{l}{100-l}=6.0\times \frac{40.0}{60.0}=4.0\Omega$$

Good practice:

- Choose R so the null point lies near the middle of the wire, where the percentage error is least
- Swap X and R and average the two results to cancel end errors

An everyday example. An electronic weighing scale uses strain gauges in a bridge circuit; a load unbalances the bridge, and the tiny imbalance shows up as weight.

The substance. The balance condition does not depend on the galvanometer's resistance or the cell's emf, which is why the bridge is so accurate.

**A potentiometer is a long uniform wire carrying a steady current, so the potential drop along it is proportional to length; balancing a cell against a length l gives $\epsilon =kl$, so $\frac{{\epsilon }_1}{{\epsilon }_2}=\frac{l_1}{l_2}$ and $r=R\left(\frac{l_1-l_2}{l_2}\right)$.

Why it beats a voltmeter. At the balance point no current flows through the cell, so the potentiometer measures its true emf. A voltmeter always draws some current and reads the terminal voltage instead.

Worked example 1 — comparing emfs.** A driver cell sets 2.0 V across a 4.0 m wire, so $k=0.50$ V m${}^{-1}$. A 1.5 V cell balances at 3.00 m and an unknown cell at 2.20 m:

$${\epsilon }_2=1.5\times \frac{2.20}{3.00}=1.10\text{ V}$$

Worked example 2 — internal resistance. A cell balances at 1.20 m on open circuit. With a $10\Omega$ resistor across it, the balance moves to 1.00 m:

$$r=10\times \frac{1.20-1.00}{1.00}=2.0\Omega$$

Conditions for a balance point:

- The driver cell's emf must exceed the emf being measured
- Positive terminals of both cells must join the same end of the wire

An everyday example. The volume knob of an old radio is a rotary potentiometer — turning it slides a contact along a resistive track to pick off a fraction of the voltage.

The substance. A longer potentiometer wire gives a smaller potential gradient, spreading the balance over more length and so improving sensitivity.