Question 76: Current Electricity

Question

Two resistances $R_{1}=X \Omega$ and $R_{2}=1 \Omega$ 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 $0.2 \mathrm{~mm}$ at $A$ to $1 \mathrm{~mm}$ at $B$. A galvanometer $(\mathrm{G})$ connected to the center of the wire, $50 \mathrm{~cm}$ from each end along its axis, shows zero deflection when $A$ and $B$ are connected to a battery. The value of $X$ is ____________.

Accepted Answer

According to the given condition wire is having different cross-section at two junction. (Frustum shaped conducter)
$$
R=ho \frac{I}{\pi a b}
$$
For the shown conductor in the diagram.
$$
r=\frac{a+b}{2}=\frac{0.2+1}{2}=0.6
$$ .........(i)
As
$$
\mathrm{R}=\frac{ho l}{\mathrm{~A}}
$$ ...........(ii)
Hence, Resistence of left $50 \mathrm{~cm}$ wire
$$
=\frac{ho 	imes 0.5 	imes 10^6}{\pi 	imes 0.2 	imes 0.6}
$$
Resistence of Right $50 \mathrm{~cm}$ wire
$$
=\frac{ho 	imes 0.5 	imes 10^6}{\pi 	imes 0.6 	imes 1}
$$
For wheatstone balanced condition
$

\frac{R_1}{P}  =\frac{R_2}{Q} \quad\left(R_1=Xight) $
$$ \Rightarrow $$ $
\frac{(X) 	imes \pi 	imes 0.2 	imes 0.6}{ho 	imes 0.5 	imes 10^6} =\frac{(1 \pi) 	imes 0.6 	imes 1}{ho 	imes 0.5 	imes 10^6} $
$$ \Rightarrow $$ $
\frac{(X) 	imes \pi 	imes 0.12}{ho 	imes 0.5 	imes 10^6} =\frac{\pi 	imes 0.6 	imes 1}{ho 	imes 0.5 	imes 10^6}$
$$ \Rightarrow $$ $X =5$

Current Electricity

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