A uniform potentiometer wire carries a steady current. A reference cell of e.m.f. 1.20 V balances at 60.0 cm and an unknown cell balances at 40 cm, both measured from the same end with unchanged driver current. At balance the detector current is zero. (a) Find the unknown e.m.f. (b) Explain why its internal resistance does not affect the balance. (c) Explain the error if driver current increases only before measuring the unknown cell.

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Unknown e.m.f. 0.8 V; no test-cell current means no internal voltage drop; increasing driver current makes the inferred unknown e.m.f. too small. A uniform wire carrying constant current has resistance and voltage drop proportional to length. At null balance the potential difference along the selected length equals the test cell's e.m.f., so the same voltage gradient applies to both cells. The reference establishes a gradient of 1.20/60.0 = 0.0200 V/cm. Thus the unknown e.m.f. is 0.0200 × 40 = 0.8 V, or equivalently 1.20 × 40/60.0. At balance there is no current through the test cell and detector branch, so the test cell's internal drop Ir is zero. This is different from the driver circuit, where current must still flow to maintain the wire's voltage gradient. If driver current increases for the unknown measurement, voltage gradient increases and that cell balances at a shorter length. Using the old smaller gradient with this shortened length underestimates its e.m.f. Keep driver current and wire temperature stable or recalibrate with the reference.

Comparing cells by a null measurement 1

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