A resistance of $2\,\Omega $ is connected across one gap of a meter-bridge and unknown resistance, greater than $2\,\Omega $ , is connected a cross the other gap. When these resistances are interchanged, the balance point shifts by $20\ cm$ , neglecting any end correction, the unknown resistance is ................ $\Omega$
Diffcult
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$\frac{x}{2} = \frac{\ell }{{1 - \ell }}$
$\frac{2}{x}=\frac{\ell+0.2}{1-(\ell-0.2)}=\frac{\ell+0.2}{0.8-\ell}$
$\frac{\ell(\ell+0.2)}{(1-\ell)(0.8-\ell)}=1$
$\ell^{2}+0.2 \ell=0.8+\ell^{2}-0.8 \,\ell-\ell$
$2\, \ell=0.8$
$\ell=0.4 \mathrm{\,m}$
$\frac{x}{2}=\frac{0.4}{0.6} \Rightarrow x=\frac{4}{3}$
$\frac{2}{x}=\frac{\ell-0.2}{1-(\ell-0.2)}=\frac{\ell-0.2}{1.2-\ell}$
$1=\frac{\ell(\ell-0.2)}{(1-\ell)(1.2-\ell)}$
$\ell-0.2\, \ell=1.2+\ell^{2}-2.2\, \ell$
$2 \ell=1.2$
$\ell=0.6$
$\Rightarrow x=\frac{2 \times 0.6}{0.4}=3\, \Omega$
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