Question

Given n resistors each of resistance R, how will you combine them to get the (i) maximum (ii) minimum effective resistance? What is the ratio of the maximum to minimum resistance?

Answer 1

The two fundamental ways to combine resistors are in series and in parallel. These two configurations yield the maximum and minimum possible resistances, respectively.

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(i) Maximum Effective Resistance

To obtain the maximum possible effective resistance, you must connect all n resistors in series.

How to combine them:
In a series combination, the resistors are connected end-to-end, so the current flows through each resistor one after the other.

Calculation:
The formula for the equivalent resistance ($R{series}$) of resistors in series is the sum of their individual resistances:
$R{series} = R_1 + R_2 + ... + R_n$

Since each of the n resistors has a resistance of R, the calculation is:
$R_{max} = R + R + ... + R$ (n times)

Therefore, the maximum effective resistance is:
$R_{max} = nR$

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(ii) Minimum Effective Resistance

To obtain the minimum possible effective resistance, you must connect all n resistors in parallel.

How to combine them:
In a parallel combination, all the resistors are connected across the same two points, providing multiple paths for the current to flow.

Calculation:
The formula for the equivalent resistance ($R{parallel}$) of resistors in parallel is given by the reciprocal of the sum of the reciprocals of their individual resistances:
$1/R{parallel} = 1/R_1 + 1/R_2 + ... + 1/R_n$

Since each of the n resistors has a resistance of R, the calculation is:
$1/R_{min} = 1/R + 1/R + ... + 1/R$ (n times)

$1/R_{min} = n/R$

To find $R{min}$, we take the reciprocal of this result:
$R{min} = R/n$

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Ratio of the Maximum to Minimum Resistance

Now, we find the ratio of the maximum resistance ($R{max}$) to the minimum resistance ($R{min}$).

Values:
Maximum Resistance, $R_{max} = nR$
Minimum Resistance, $R_{min} = R/n$

Ratio Calculation:
Ratio = $R{max} / R{min}$

Ratio = $(nR) / (R/n)$

To simplify, we can multiply by the reciprocal of the denominator:
Ratio = $nR \times (n/R)$

The R terms in the numerator and denominator cancel out:
Ratio = $n \times n$

Ratio = $n^2$

So, the ratio of the maximum to minimum resistance is $n^2 : 1$.

Summary

| Configuration | How to Combine | Effective Resistance |
| :--- | :--- | :--- |
| (i) Maximum | All n resistors in Series | $R{max} = nR$ |
| (ii) Minimum | All n resistors in Parallel | $R{min} = R/n$ |
| Ratio | $R{max} / R{min}$ | $n^2$ |