Two resistances and are connected to a wire of uniform resistivity, as shown in the figure. The radius of the wire varies linearly along its axis from at A to at B. A galvanometer (G) connected to the center of the wire, from each end along its axis

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 _______. [2022]

Smart Achievers · Accepted Answer

(5)

$For a frustum, R = \frac{ρ ℓ}{π a b}$ $and$ $r_{m} = \frac{a + b}{2}$

$So, R_{A C} = \frac{ρ (ℓ / 2)}{π (a r_{m})}$

$R_{C B} = \frac{ρ (ℓ / 2)}{π r_{m} b}$

By Wheatstone bridge principle,

$\frac{R_{1}}{R_{2}} = \frac{R_{A C}}{R_{C B}}$ $\Rightarrow \frac{R_{1}}{1} = \frac{b}{a} = \frac{1}{0.2} = 5$

$∴$ $R_{1} = 5 Ω$

Solution

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Solution

Ans. (5) For a frustum, R=ρℓπab and rm=a+b2 So, RAC=ρ(ℓ/2)π(arm) RCB=ρ(ℓ/2)πrmb By Wheatstone bridge principle, R1R2=RACRCB⇒R11=ba=10.2=5 ∴ R1=5 Ω

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