When two identical batteries of internal resistance $1 \Omega$ each are connected in series across a resistor $\mathrm{R}$, the rate of heat produced in $R$ is $J_1$. When the same batteries are connected in parallel across $R$, the rate is $\mathrm{J}_2$. If $\mathrm{J}_1=2.25 \mathrm{~J}_2$ then the value of $\mathrm{R}$ in $\Omega$ is
IIT 2010, Advanced
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$ \mathrm{~J}_1=\left(\frac{2 \mathrm{E}}{\mathrm{R}+2}\right)^2 \mathrm{R} $
$ \mathrm{J}_2=\left(\frac{\mathrm{E}}{\mathrm{R}+1 / 2}\right)^2 \mathrm{R} \text { since } \mathrm{J}_1 / \mathrm{J}_2=2.25 $
$ \mathrm{R}=4 \Omega .$
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