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

Answer 1

Answer: (C)

$0.25m$

For the given wire :$dR=C\frac{dl}{\sqrt{l}}$,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 $\int dR=c\int \frac{dl}{\sqrt{l}}$

$\therefore R_1=c{\int }_0^l{l}^{-1/2}dl=C.2.\sqrt{l}$

$R_2=c{\int }_l^1{l}^{-1/2}dl=C.(2-2\sqrt{l})$

Putting $R_1=R_2$

$C2\sqrt{l}=C(2-2\sqrt{l})$

$\therefore 2\sqrt{l}=1$

$\sqrt{l}=\frac{1}{2}$

$l=\frac{1}{4}m$

$\Rightarrow 0.25m$