Question

A set of n-identical resistors, each of resistance R ohm when connected in series have an effective resistance of X ohm and when the resistors are connected in parallel the effective resistance is Y ohm. Find the relation between R , X and Y ?

Answer 1

Let's break this down step-by-step to find the relationship between R, X, and Y.

1. Resistors in Series

When resistors are connected in series, their total effective resistance is the sum of their individual resistances.

• You have n identical resistors.
• The resistance of each resistor is R.

The effective resistance, which is given as X, is:
$X = R + R + R + ...$ (n times)
$X = n \times R$

Let's call this Equation (1):
$X = nR$

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2. Resistors in Parallel

When resistors are connected in parallel, the reciprocal of the total effective resistance is the sum of the reciprocals of their individual resistances.

• The effective resistance in this case is given as Y.

So, the formula is:
$1/Y = 1/R + 1/R + 1/R + ...$ (n times)
$1/Y = n \times (1/R)$
$1/Y = n/R$

To find Y, we can take the reciprocal of both sides:
$Y = R/n$

Let's call this Equation (2):
$Y = R/n$

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3. Finding the Relation between R, X, and Y

Now we have two simple equations:
1. $X = nR$
2. $Y = R/n$

Our goal is to find a relationship that only involves R, X, and Y, which means we need to eliminate the variable n. A very simple way to do this is to multiply Equation (1) by Equation (2).

$X \times Y = (nR) \times (R/n)$

The 'n' in the numerator and the 'n' in the denominator will cancel each other out:

$X \times Y = (n \times R \times R) / n$
$XY = R \times R$
$XY = R^2$

Conclusion

The relation between R, X, and Y is:

$R^2 = XY$

You can also express this as:

$R = \sqrt{XY}$

This means that the resistance of a single resistor (R) is the geometric mean of the total series resistance (X) and the total parallel resistance (Y).