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 $\Omega$

(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,

$\frac{1}{R\text{'}}=\frac{1}{R}+\frac{1}{R}=\frac{2}{R}=\frac{2}{30}=\frac{1}{15}$
$\Rightarrow R\text{'}=15\Omega$

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

R_eq = R' + R = 15 + 30 = 45 $\Omega$

(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 $\Omega$

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

$\frac{1}{R_{eq}}=\frac{1}{R\text{'}}+\frac{1}{R}=\frac{1}{60}+\frac{1}{30}=\frac{3}{60}=\frac{1}{20}$
$\Rightarrow R_{eq}=20\Omega$