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

Question

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

Aicharya · Accepted Answer

Left part of the resistors are connected in parallel, hence $R_{1} = \frac{a m}{3}$ .

Similarly, for the right part of the resistors we can write, $R_{2} = \frac{a}{2 m}$ .

Therefore, equivalent resistance will be,

$R = \frac{a m}{3} + \frac{a}{2 m}$

Differentiating both sides w.r.t $m$ to observe critical points, we get $\frac{d R}{d m} = \frac{a}{3} + \frac{a (- 1)}{2 m^{2}} = 0$

$\Rightarrow \frac{a}{3} = \frac{a}{2 m^{2}}$ $\Rightarrow m = \sqrt{\frac{3}{2}}$

Thus, $x = 3$ .

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