Metre bridge theory

Place the unknown R in the left gap and known S in the right gap. The wire runs from A at 0 cm to B at 100 cm. A galvanometer joins the gap junction to a movable jockey J on the wire. Define l as AJ, measured from the unknown-resistor end. A formula remembered without this orientation can invert the resistance ratio.

At balance no current flows through the galvanometer, so its two terminals are at equal potential. The gap resistors form one potential divider and the two wire segments form another. Equating the fractional voltage drops gives R/S equal to the ratio of the two wire resistances. This equality uses the null condition, not the assumption that all currents in the circuit are equal.

For a wire of uniform resistivity and cross-sectional area, each segment resistance is proportional to length. The common factor cancels in the ratio, giving R/S = l/(L − l). Both lengths must use the same unit. If S = R, the model balance lies at the midpoint. Wire nonuniformity and resistance at the ends disturb this ideal relation.

Rearrange the bridge equation as (100.0 − l)/l = S/R. Plot known resistance S in Ω horizontally and the dimensionless ratio y = (100.0 − l)/l vertically. The slope has unit Ω⁻¹ and equals 1/R; R = 1/m. A free intercept helps reveal systematic departures but cannot correct every end effect.

For a fixed absolute length uncertainty, the relative resistance error from l is approximately L u(l)/[l(L − l)]. This is smallest near the midpoint and grows towards either end. Choose S similar to R in a physical experiment so balances usually lie near the centre; an exploratory range can demonstrate why end positions are less suitable. The available 1 Ω setting gives a balance near 83.3 cm and should not be treated as a preferred physical setting.

In physical work exchange R and S, measure the new balance from the same geometric zero, and apply the corresponding reversed formula R = S(L − l_reversed)/l_reversed. Compare both determinations; interchange can diagnose and reduce some end effects, without removing arbitrary wire nonuniformity. Include uncertainty of S, null detection and l, and inspect drift. Ordinary least squares here treats S as exact and does not weight the transformed length uncertainties.