Potentiometer

PHYMeasuring Instruments

A potentiometer measures potential difference and EMF by balancing an unknown against a uniform potential gradient along a long wire; at balance the device draws no current, so it reads true EMF without loading the cell. ISC uses it to compare two EMFs, to find the internal resistance of a cell, and to compute the potential gradient and balancing length.

The key idea is that balancing length is directly proportional to the potential being balanced.

Potential gradient

k = \dfrac{V}{L}

k

potential gradient (V/m or V/cm),

V

potential difference across the whole wire,

L

total wire length. Balanced PD on a length

l

V_l = k\,l

Comparing two EMFs

\dfrac{E_1}{E_2} = \dfrac{l_1}{l_2}

l_1, l_2

balancing lengths for cells of EMF

E_1, E_2

under the same potential gradient. EMF is proportional to balancing length.

Internal resistance from a potentiometer

r = R \left( \dfrac{l_1 - l_2}{l_2} \right)

l_1

balancing length on open circuit (EMF),

l_2

with shunt

R

across the cell (terminal voltage),

R

the external resistance. Uses

r = R(E-V)/V

EMF from balancing length

E = k \, l

E

EMF balanced (V),

k

potential gradient (V/cm),

l

balancing length (cm). Rearranges to find

l = E/k

for a known EMF.

At the balance point the galvanometer shows no deflection, so the test cell supplies zero current and the potentiometer reads true EMF (no $Ir$ drop).
The driver (auxiliary) cell must have EMF greater than any EMF being measured, otherwise no balance point exists on the wire.
Potential gradient $k=V/L$ must be uniform — the wire needs constant cross-section and a steady driving current.
A potentiometer is more accurate than a voltmeter because a voltmeter always draws some current and reads terminal voltage, not EMF.
For internal resistance, $l_1$ corresponds to open-circuit EMF and $l_2$ to terminal voltage with the cell loaded by $R$ .
A longer wire (smaller $k$ ) gives greater sensitivity because a given EMF balances over a longer, more readable length.
Balancing lengths can be used directly as proxies for voltages since both are proportional to $k$ .

Where the marks go

Inverting the internal-resistance formula — it is $r=R(l_1-l_2)/l_2$ , with the shorter loaded length $l_2$ in the denominator.
Forgetting the driver-cell EMF must exceed the measured EMF, otherwise the question has no balance point.
Mixing cm and m inconsistently between gradient and length so units of $k$ and $l$ do not cancel.
Treating the potentiometer like a voltmeter and assuming it reads terminal voltage rather than true EMF at balance.

How the board asks it

Numericalinternal resistance of a cell
In a potentiometer experiment the EMF of a cell balances at a length $l_1 = 560\,\text{cm}$ . When a resistance $R = 10\,\Omega$ is connected across the cell, the balancing length falls to $l_2 = 500\,\text{cm}$ . Calculate the internal resistance of the cell.
Numericalcomparison of two EMFs and potential gradient
A potentiometer wire of length $10\,\text{m}$ carries a steady current. Two cells are balanced separately at $250\,\text{cm}$ and $400\,\text{cm}$ . Find the ratio of their EMFs. If the second cell has EMF $1.5\,\text{V}$ , calculate the potential gradient of the wire in $\text{V m}^{-1}$ .
Give reasonszero current at balance reads true EMF
Explain why a potentiometer measures the EMF of a cell more accurately than a voltmeter does.
Diagram / graphcomparison of two EMFs setup
Draw a labelled circuit diagram of a potentiometer used to compare the EMFs of two cells, and state the condition the driver cell must satisfy for a balance point to be obtained on the wire.
Give reasonssmaller gradient gives greater sensitivity
Explain why the sensitivity of a potentiometer increases when the length of its wire is increased for a given driving current.
Define / statepotential gradient
Define the potential gradient of a potentiometer wire and state the conditions necessary for it to remain uniform along the length of the wire.

Practise this topic

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Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.