Question

A set of n identical resistors each of resistance R are connected in series and the effective resistance is found to be X. When these n resistors are connected in parallel the effective resistance is found to be Y find the ratio of X and Y

Answer 1

In series
nR = X

In parellel
R/n = Y

Ratio of X and Y = nR / (R/n) = n²

X : Y = n² : 1

Answer 2

The ratio of X and Y is the square of number of identical resistors.

Explanation:

Let us consider the resistance of each resistors be R.  

The effective resistance in series,

$R_{S}=R_{1}+R_{2}+R_{3}+R_{4}+\cdots \cdots \cdots+R_{n}=\sum_{1}^{n} R=X$

As all the n resistors have same resistance, then the effective resistance in series becomes,

$X=n R \rightarrow(1)$

The effective resistance in parallel connection is,

$\frac{1}{R_{p}}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}+\cdots \cdots+\frac{1}{R_{n}}=\frac{1}{Y}=\frac{n}{R}$

As all the n resistors have same resistance, then,

$Y=\frac{R}{n} \rightarrow(2)$

The ratio of X and Y is,

$X : Y=\frac{X}{Y}$

So divide equation (2) with equation (1),

$X : Y=\frac{n R}{(R / n)}=\frac{n R \times n}{R}=n^{2}$

$X : Y= n^{2} : 1$