Two resistances R 1 = X Ω and R 2 = 1 Ω are connected to a wire A B of uniform resistivity, as shown in the figure. The radius of the wire varies linearly along its axis from 0 . 2 mm at A to 1 mm at

Question

Two resistances R 1 = X Ω and R 2 = 1 Ω are connected to a wire A B of uniform resistivity, as shown in the figure. The radius of the wire varies linearly along its axis from 0 . 2 mm at A to 1 mm at B . A galvanometer G connected to the center of the wire, 50 cm from each end along its axis, shows zero deflection when A and B are connected to a battery. The value of X is _______.

TestHub · Accepted Answer

Resistance of frustum shaped conductor shown is
$R = ρ \frac{I}{π a b}$
For the shown conductor in the diagram.
$r = \frac{a + b}{2} = \frac{0.2 + 1}{2} = 0.6 mm$
Thus, the resistance of left half is $P = \frac{ρ \times 0.5 \times 10^{6}}{π \times 0.2 \times 0.6}$
And the resistance of right half is $Q = \frac{ρ \times 0.5 \times 10^{6}}{π \times 0.6 \times 1}$
For Wheatstone to be balanced
$\frac{R_{1}}{P} = \frac{R_{2}}{Q}$
$\Rightarrow \frac{X π \times 0.2 \times 0.6}{ρ \times 0.5 \times 10^{6}} = \frac{1 π \times 0.6 \times 1}{ρ \times 0.5 \times 10^{6}}$
$\Rightarrow X = 5$

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