Chapter 21, Problem 1OQ

If the terminals of a battery with zero internal resistance are connected across two identical resistors in series, the total power delivered by the battery is 8.00 W. If the same battery is connected across the same resistors in parallel, what is the total power delivered by the battery? (a) 16.0 W (b) 32.0 W (c) 2.00 W (d) 4.00 W (e) none of those answers

To determine

The total power delivered by the battery if it is connected across the two identical resistors in parallel.

Answer to Problem 1OQ

The total power delivered by the battery if it is connected across the two identical resistors in parallel is $\underset{\_}{\left(\text{b}\right)32.0\text{W}}$.

Explanation of Solution

Write the expression for the current supplied by the battery when the two identical resistors are in series,

    $I=\frac{\Delta V}{2R}$        (I)

Here, $I$ is the current supplied by the battery, $\Delta V$ is the potential difference and $R$ is the resistance.

Write the expression for the total power delivered by the battery in series,

    $P_s=\left(\Delta V\right)I$        (II)

Here, $P_s$ is the total power delivered by the battery in series.

Substitute (I) in (II),

    $P_s=\frac{{\left(\Delta V\right)}^2}{2R}$        (III)

The two identical resistors connected in parallel, the potential difference across each resistor is $\Delta V$ .

Write the expression for the power delivered to each resistor in parallel,

    $P_1=\frac{{\left(\Delta V\right)}^2}{R}$        (IV)

Here, $P_1$ is the power delivered to each resistor in parallel.

Write the expression for the total power delivered by the battery in parallel,

    $P_p=2P_1$        (V)

Here, $P_p$ is the total power delivered by the battery in parallel.

Substitute (IV) in (V),

    $\begin{array}{c}{P}_p=2\left[\frac{{\left(\Delta V\right)}^2}{R}\right]\\ =4\left[\frac{{\left(\Delta V\right)}^2}{2R}\right]\\ =4P_s\end{array}$        (VI)

Conclusion:

Substitute $8.00\text{W}$ for $P_s$ in the above equation,

    $\begin{array}{c}{P}_p=4\left(8.00\text{W}\right)\\ =32.0\text{W}\end{array}$

Therefore, the total power delivered by the battery if it is connected across the two identical resistors in parallel is $\underset{\_}{\left(\text{b}\right)32.0\text{W}}$.