Question 130 of 380

In the given circuit 'a' is an arbitrary constant. The value of $m$ for which the equivalent circuit resistance is minimum, will be $\sqrt{\dfrac{x}{2}}$. The value of $x$ is _____ .

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Answer: 3

Left part of the resistors are connected in parallel, hence ${R}_{1}=\dfrac{am}{3}$.
Similarly, for the right part of the resistors we can write, ${R}_{2}=\dfrac{a}{2m}$.
Therefore, equivalent resistance will be,
$R=\dfrac{am}{3}+\dfrac{a}{2m}$
Differentiating both sides w.r.t $m$ to observe critical points, we get $\dfrac{dR}{dm}=\dfrac{a}{3}+\dfrac{a\left(-1\right)}{2{m}^{2}}=0$
$\Rightarrow \dfrac{a}{3}=\dfrac{a}{2{m}^{2}}$ $\Rightarrow m=\sqrt{\dfrac{3}{2}}$
Thus, $x=3$.

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