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Conductivity and Resistivity

The equivalen...

Question

The equivalent resistarce of $r_{1}$ & $r_{2}$ when connected in series in $R_{1}$ and that when they are connected in parallel $R_{1}$ .
Then the ratio $\frac{R_{1}}{R_{2}}$ is

A

\frac{r_{1}}{r_{2}}

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B

\frac{r_{1} + r_{2}}{r_{1} r_{2}}

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C

\frac{(r_{1} + r_{2})^{2}}{r_{1} r_{2}}

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D

\frac{r_{1} r_{2}}{2 r_{1} + r_{2}}

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Solution

The correct option is D $\frac{(r_{1} + r_{2})^{2}}{r_{1} r_{2}}$
For series
$R_{1} = r_{1} + r_{2}$ (equivalent resistance)

for parallel, $\frac{1}{R} = \frac{1}{r_{1}} + \frac{1}{r_{2}}$

$R_{2} = \frac{r_{1} r_{2}}{r_{1} + r_{2}}$ (equivalent resistance)

$∴ \frac{R_{1}}{R_{2}} = \frac{(r_{1} + r_{2})^{2}}{r_{1} r_{2}}$

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