A wire AB of length 1 m has a uniform resistivity but a radius that increases linearly along its axis from 0.2 mm at end A to 1 mm at end B. A resistance R1 = X ohm is joined at A and R2 = 1 ohm is joined at B, and the combination is driven by a battery. A galvanometer is tapped to the wir

Question

A wire AB of length 1 m has a uniform resistivity but a radius that increases linearly along its axis from 0.2 mm at end A to 1 mm at end B. A resistance R1 = X ohm is joined at A and R2 = 1 ohm is joined at B, and the combination is driven by a battery. A galvanometer is tapped to the wire at its axial midpoint C, which lies 50 cm from each end. If the galvanometer reads zero, find the value of X.

Accepted Answer

5. Balance condition for a meter-bridge-type null gives X/R(AC) = R2/R(CB), so X = R(AC)/R(CB) (with R2 = 1). The wire resistance per unit length scales as 1/r². Since the segment integral of dx/r² reduces to (1/r_start - 1/r_end) times a common factor, only the endpoint radii matter. The radius at the midpoint C is the average, 0.6 mm.

Solution

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