Question

In a meter bridge, the wire of length $1m$ 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 $R_2$
$\therefore \frac{R^{\prime }}{R^{\prime }}=\frac{R_1}{R_2}$ or $R_1=R_2$ By balanced
WSB concept.
Now $\int dR=c\int \frac{d\ell }{\sqrt{\ell }}$
$\therefore R_1=C{\int }_0^{\ell }{\ell }^{-1/2}d\ell =C.2.\sqrt{\ell }$
$R_2=C{\int }_{\ell }^1{\ell }^{-1/2}d\ell =C.\left(2-2\sqrt{\ell }\right)$
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. $\ell =\frac{1}{4}m\Rightarrow 0.25m$