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

Step 1: Given data

1. N number of resistors of resistance R.

Step 2: Combination of resistance

1. Resistance is a property of the material (resistor) that resists the current flow through it.
2. There have two types of combinations one is the series combination and another is the parallel combination.
3. In a series combination of resistance R,$totalresis\tan ce\left(R_s\right)=numberofresis\tan ce\left(N\right)\times valueofoneresis\tan ce\left(R\right)$, i.e, $R_s=NR.$
4. In a parallel combination of resistance R, $totalresis\tan ce\left(R_p\right)=\frac{totalnumberofresis\tan ce\left(N\right)}{valueofoneresis\tan ce\left(R\right)}$, i.e, $R_p=\frac{N}{R}.$

Step 3: Finding the ratio

It is clear that in a series combination of resistance, the equivalent resistance is maximum $\left(NR\right)$,

and in a series combination, the equivalent resistance is minimum $\frac{N}{R}$.

So, the ratio of minimum and maximum resistance is,

$$\begin{gathered} n=\frac{{\displaystyle \frac{N}{R}}}{NR}=\frac{1}{R^2} \\ orn=\frac{1}{R^2}. \end{gathered}$$

Therefore, the ratio of the minimum to maximum resistance is $1:R^2$.