Uses of Potentiometer

Because the potential gradient K stays fixed once the driving cell, the wire and the rheostat setting are all fixed, an unknown potential difference can be measured simply by finding the LENGTH of wire at which it exactly balances -- the null point, where a connected galvanometer shows zero deflection. This one idea, applied slightly differently each time, underlies every use of the potentiometer described below.

A) To compare the emf of cells

Method I -- connecting the cells individually. A potentiometer circuit is set up with a driving battery of emf $\varepsilon$, a key K and a rheostat across the wire AB, wired so that A is at higher potential than B. The two cells to be compared, of emf $\varepsilon_1$ and $\varepsilon_2$, both have their POSITIVE terminals connected to point A, while their negative terminals go to the two outer terminals of a two-way key $K_1K_2$; the central terminal of this key connects to a galvanometer, which in turn connects to a jockey touching the wire. With key K closed and $K_1$ closed (while $K_2$ stays open), only the first cell $\varepsilon_1$ is in circuit with the galvanometer, and its null point is found at some length $l_1$ from A, so that

where k is the potential gradient of the wire. Then $K_1$ is opened and $K_2$ closed instead, bringing $\varepsilon_2$ into the same circuit; its own null point is found at length $l_2$ from A, so that

Dividing the two equations cancels the (unknown) potential gradient k entirely, giving

so the two emfs can be compared purely from the two balancing LENGTHS -- and if either emf is independently known, the other follows immediately.

Method II -- the sum and difference method. When two cells are joined so the NEGATIVE terminal of the first meets the POSITIVE terminal of the second (exactly as cells are strung together in an ordinary battery), their emfs act in the same sense and add: the pair behaves as one effective source of emf $(\varepsilon_1+\varepsilon_2)$ -- this is called the SUM method (note that this is NOT the same thing as a parallel combination of cells). When instead the two cells are joined with their LIKE terminals together (both positive terminals tied together, or both negative terminals tied together), their emfs oppose and the pair behaves as one effective source of emf $(\varepsilon_1-\varepsilon_2)$, taking $\varepsilon_1$ as the larger of the two -- this is the DIFFERENCE method.

With both cells wired through four keys $K_1,K_2,K_3,K_4$ so that closing $K_1,K_3$ selects the sum configuration and closing $K_2,K_4$ selects the difference configuration, the SAME potentiometer wire and galvanometer are used to find a null point for each in turn. Let $l_1$ be the balancing length for the sum mode, so

and $l_2$ the balancing length for the difference mode, so

Dividing these two equations gives

and applying componendo-and-dividendo to this ratio isolates the emf ratio directly:

giving a second, independent way to compare the two emfs.

B) To find the internal resistance (r) of a cell

The set-up again uses a potentiometer wire AB in series with a driving cell of emf $\varepsilon$, a key $K_1$ and a rheostat, with A at higher potential than B. The cell whose internal resistance $r_1$ is to be found (its own emf $\varepsilon_1$) is connected to the potentiometer wire through a galvanometer G and jockey J, and a resistance box R is connected across this same cell $\varepsilon_1$ through a second key $K_2$.

With $K_1$ closed and $K_2$ open, the circuit is simply the driving cell $\varepsilon$, the test cell $\varepsilon_1$ and the potentiometer wire; the null point gives a length $l_1$ corresponding to the FULL emf of the test cell:

Now both $K_1$ and $K_2$ are closed, bringing the resistance box R into the circuit as an external load on the test cell; some resistance R is set on the box, and the (new, shorter) null point gives a length $l_2$. Since the test cell is now delivering current I through R (with r its own internal resistance), this second balance length corresponds only to the TERMINAL potential difference $V=IR$ across the loaded cell, not its full emf:

Applying Kirchhoff's voltage law to the test cell's own loop, $\varepsilon_1 = IR+Ir$ while $V=IR$, gives

and combining this with $\varepsilon_1/V = l_1/l_2$ from above and solving for r gives the internal-resistance formula

used numerically in Ex. 9.6, where a 1.5 V cell's balance length shifts from 76.3 cm (open circuit) to 64.8 cm once loaded by a 9.5 \Omega resistor, giving an internal resistance of about 1.69 \Omega.

C) Applications of the potentiometer …