Question

Suppose you have three resistors, each of value 30 Ω. List all the different resistances you can obtain using them.

Answer 1

(a) When the three resistors are connected in series:-

The resultant resistance, R_eq = R + R + R = 90 Ω

(b) When the three resistors are connected in parallel:-

The resultant resistance of the combination,

\[\frac{1}{R_{eq}} = \frac{1}{R} + \frac{1}{R} + \frac{1}{R} = \frac{3}{R} = \frac{3}{30} = \frac{1}{10}\]

\[\Rightarrow R_{eq} = 10\Omega\]

(c) When two of the resistors are connected in parallel and this combination is connected in series with the third resistor:-

Let R' be the resultant resistance of the two resistors connected in parallel to each other. Therefore,

\[\frac{1}{R'} = \frac{1}{R} + \frac{1}{R} = \frac{2}{R} = \frac{2}{30} = \frac{1}{15}\]

\[\Rightarrow R' = 15 \Omega\]

Now, the net resistance of the combination of the resistors,

R_eq = R' + R = 15 + 30 = 45 Ω

(d) When two of the resistors are connected in series and the combination is connected to the third resistor in parallel:-

Let R' be the resultant resistance of the series in combination. Therefore,

R' = R + R = 30 + 30 = 60 Ω

Now, let the net resultant of the combination be R_eq. So,

\[\frac{1}{R_{eq}} = \frac{1}{R'} + \frac{1}{R} = \frac{1}{60} + \frac{1}{30} = \frac{3}{60} = \frac{1}{20}\]

\[\Rightarrow R_{eq} = 20 \Omega\]