You are given n resistors each of resistance r. They are first connected to get the minimum possible resistance. In the second case, these are again connected differently to get the maximum possible resistance. Calculate the ratio between minimum and maximum values of resistance so obtained.
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For minimum possible resistance, the resistors should be connected in parallel
$\frac{1}{\text{r}_\text{p}}=\frac{1}{\text{r}}+\frac{1}{\text{r}}+\dots\text{n}\text{ times}$
$\frac{1}{\text{r}_\text{p}}=\frac{\text{n}}{\text{r}}$
$\text{r}_\text{p}=\frac{\text{r}}{\text{n}}$
For maximum possible resistance, the resistors should be connected in series,
$\text{r}_\text{s}=\text{r}+\text{r}+\dots\text{n times}$
$\text{r}_\text{s}=\text{nr}$
$\text{Ratio}\frac{\text{r}_\text{p}}{\text{r}_\text{s}}=\frac{\text{r}}{\text{n}}\times\frac{1}{\text{nr}}=\frac{1}{\text{n}^2}$
$\frac{1}{\text{r}_\text{p}}=\frac{1}{\text{r}}+\frac{1}{\text{r}}+\dots\text{n}\text{ times}$
$\frac{1}{\text{r}_\text{p}}=\frac{\text{n}}{\text{r}}$
$\text{r}_\text{p}=\frac{\text{r}}{\text{n}}$
For maximum possible resistance, the resistors should be connected in series,
$\text{r}_\text{s}=\text{r}+\text{r}+\dots\text{n times}$
$\text{r}_\text{s}=\text{nr}$
$\text{Ratio}\frac{\text{r}_\text{p}}{\text{r}_\text{s}}=\frac{\text{r}}{\text{n}}\times\frac{1}{\text{nr}}=\frac{1}{\text{n}^2}$
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