JEE Advanced 2022 · Paper 2 · Physics Q3

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PhysicsInteger+3 / -1Question 21 of 54

Two resistances $R_1 = X\ \Omega$ and $R_2 = 1\ \Omega$ are connected to a wire $AB$ 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 _____.

A wire AB (1 m long, tapering from radius 0.2 mm at A to 1 mm at B) is connected across a battery. R1 runs from A up to an apex and R2 from the apex down to B; a galvanometer G joins the apex to the midpoint of the wire (a bridge arrangement).

Your answer

Answer and worked solution

Official answer: 5

Resistance of the tapered wire. The radius is $r(x)=0.2+0.8x$ mm, with $x$ in metres from $A$ . A slice has resistance $\dfrac{\rho\,dx}{\pi r^2}$ , and
$\int\frac{dx}{(0.2+0.8x)^2}=-\frac{1}{0.8(0.2+0.8x)}.$

$R_{AM}\propto\frac1{0.8}\left(\frac1{0.2}-\frac1{0.6}\right)=4.17$ .
$R_{MB}\propto\frac1{0.8}\left(\frac1{0.6}-\frac11\right)=0.83$ .

So $R_{AM}:R_{MB}=5:1$ .

Balanced bridge. With no current in G,
$\frac{R_1}{R_2}=\frac{R_{AM}}{R_{MB}}=5\implies X=5.$

Topics:Wheatstone bridge and metre bridge Ohm's law and resistivity

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