Question

$N$ Resistors each of resistance $R$ are first combined to get minimum possible resistance and then combined to get maximum possible resistance. The ratio of the minimum to maximum resistance is ?

Answer 1

Hint:In a series circuit, all elements are connected end-to-end, creating a continuous direction for current flow. In a parallel circuit, all elements are wired over each other, creating precisely two sets of electrically common points.

Complete step by step answer:
We know that , when we combine resistance in parallel, then the resultant resistance will be reduced. Hence, the minimum possible resistance of $N$resistors will be calculated using parallel combination formula:
${R_{\max }} = R + R + R + ..... + R(N - times) \\
\Rightarrow {R_{\max }} = N.R - (i) $

Similarly, when we combine resistance in series combination, then the resultant resistance will be increased. Hence, the minimum possible resistance of $N$ resistors will be calculated using series combination formula:
$\dfrac{1}{{{R_{\min }}}} = \dfrac{1}{R} + \dfrac{1}{R} + \dfrac{1}{R} + ...... + \dfrac{1}{R}(N - times) \\
\Rightarrow \dfrac{1}{{{R_{\min }}}} = \dfrac{N}{R} \\
\Rightarrow {R_{\min }} = \dfrac{R}{N} - (ii) \\ $
Now, we divide $(ii)$ equation by $(i)$ to calculate the ratio of the minimum to maximum resistance :
$\dfrac{{{R_{\min }}}}{{{R_{\max }}}} = \dfrac{R}{N} \times \dfrac{1}{{NR}} \\
\Rightarrow \dfrac{{{R_{\min }}}}{{{R_{\max }}}} = \dfrac{{1}}{N} \times \dfrac{1}{{N}} \\
\therefore\dfrac{{{R_{\min }}}}{{{R_{\max }}}} = \dfrac{1}{{{N^2}}}$

Hence, The ratio of the minimum to maximum resistance is $\dfrac{1}{{{N^2}}}$.

Note:Since the output current of the first resistor flows through the input of the second resistor of a series circuit, the current in each resistor is the same. Many of the resistor leads on one side of the resistors are wired together in a parallel circuit, and all of the resistor leads on the other side are connected together.