Question

In a meter bridge, the wire of length $1\, m$ has a non-uniform cross-section such that, the variation $\frac{dR}{dl}$ of its resistance $R$ with length $l$ is $\frac{dR}{dl} \propto \frac{1}{\sqrt{l}}$. Two equal resistances are connected as shown in the figure. The galvanometer has zero deflection when the jockey is at point $P$. What is the length $AP$ ?

• A. 0.25 m
• B. 0.3 m
• C. 0.35 m
• D. 0.2 m

Answer 1

Answer: (A)

For the given wire : $dR = C \frac{d \ell}{\sqrt{\ell}}$ , where $C = constant$.
Let resistance of part AP is $R_1$ and PB is $R_2$
$\therefore \frac{R'}{R'} = \frac{R_{1}}{R_{2}} $ or $R_{1} =R_{2} $ By balanced
WSB concept.
Now $ \int dR = c \int \frac{d\ell}{\sqrt{\ell}} $
$ \therefore R_{1} = C \int^{\ell}_{0} \ell^{-1/2} d\ell = C.2 . \sqrt{\ell} $
$ R_{2} = C \int^{1}_{\ell} \ell^{-1/2} d\ell =C. \left(2-2\sqrt{\ell}\right) $
Putting $ R_{1} =R_{2} $
$ C_{2} \sqrt{\ell} =C \left(2-2 \sqrt{\ell}\right) $
$ \therefore 2 \sqrt{\ell} = 1 $
$ \sqrt{\ell} = \frac{1}{2} $
i.e. $ \ell = \frac{1}{4} m \Rightarrow 0.25\, m $