Three equal resistors when connected in series consume $2 W$ of power from a battery. Calculate the power consumed if these resistors are connected in parallel to the same battery.

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Solution

Step 1: Given
Three resistors of equal resistance
$P o w e r i n s e r i e s = 2 W$
Step 2: Find the power if resistances are connected in a parallel combination.
When resistances are in series
$R_{s} = R + R + R$
$= 3 R$
The total resistance in parallel.
$\frac{1}{R_{P}} = \frac{1}{R} + \frac{1}{R} + \frac{1}{R}$
$R_{P} = \frac{R}{3}$
Step 3: Calculate the power consumed in parallel combination,
We are using the same battery in both conditions so, the potential is constant.
$∵ P o w e r (P) = \frac{V^{2}}{R}$ [Where V= potential, R= Resistance]
If voltage is constant with the same battery power is inversely proportional to resistance.
$\frac{P o w e r i n s e r i e s}{P o w e r i n p a r a l l e l} = \frac{R_{p}}{R_{s}}$
$P o w e r i n p a r a l l e l c o m b i n a t i o n = \frac{P o w e r i n s e r i e s \times R_{s}}{R_{P}}$
Now, put the given value in the above formula.
$P o w e r i n p a r a l l e l c o m b i n a t i o n = \frac{2 \times 3 R}{\frac{R}{3}} = 18 watts$
Hence, power consumed in parallel combination is $18 W$ .

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