Chapter 22.2, Problem 34PP

To determine

The time taken to recharge the battery using a current of $7.5\text{A}$ and voltage of $14\text{V}$ .

Answer to Problem 34PP

  $8.2\text{h}$

Explanation of Solution

Given:

The current delivered by a battery, $I_1=55\text{A}$

The voltage supplied by a battery, $V_1=12\text{V}$

The time in which $55\text{A}$ can be delivered, $t_1=1.0\text{h}$

The current supplied during recharging, $I_r=7.5\text{A}$

The current applied during recharging, $V_r=14\text{V}$

Formula used:

The electric power $\left(P\right)$ across a circuit is given by,

  $P=IV$

Where, I is the current and V is the potential difference.

And, the energy $\left(E\right)$ supplied/delivered during time $t$ is given by,

  $\begin{array}{c}E=Pt\\ =VIt\end{array}$

Calculation:

Energy delivered by the battery during $t_1=1.0\text{h}$ will be,

  $\begin{array}{c}{E}_1=V_1{I}_1{t}_1\\ =\left(12\right)\left(55\right)\left(1\right)\\ =660\text{W}\cdot \text{h}\end{array}$

Then, the energy needed for the recharging will be,

  $E_r=1.3E_1$

Using given and calculated values in above,

  $\begin{array}{c}{V}_r{I}_r{t}_r=1.3\left(660\text{W}\cdot \text{h}\right)\\ \left(14\right)\left(7.5\right)t_r=858\\ t_r\approx 8.2\text{h}\end{array}$

Conclusion:

Hence, thetime taken to recharge the battery is $8.2\text{h}$ .