Question

A 12 V battery is charged for 20 minutes by connecting it to a source of electromotive force (e.m.f.). The battery is supplied with $7.2\times {{10}^{4}}J$ of energy in this time. How much charge flows through the battery?
$A)\text{ }5.0C$
$B)\text{ 60}C$
$C)\text{ 100}C$
$D)\text{ }6000C$

Answer 1

Hint: This problem can be solved by using the formula for the energy required by a battery to get fully charged in terms of its potential difference, the current passing through it and the time for which the current flows through the battery. Then from the current value, we can get the total charge that has passed through the battery.

Formula used: $E=VIt$
$q=It$

Complete step by step answer:
Let us write the formula for the energy supplied to the battery to charge it.
The total energy $E$ that is supplied to a battery to charge it to a potential $V$ by passing a current $I$ through it for a time $t$ is given by
$E=VIt$ --(1)
Also, the net charge $q$ that passes through the cross section of a conductor when a current $I$ flows through the conductor for a time $t$ is given by
$q=It$ --(2)
Now, let us analyze the question.
The given voltage of the battery is $V=12V$.
The energy supplied to the battery to charge it is $E=7.2\times {{10}^{4}}J$.
Let the current passing through the battery to charge it be $I$.
The time for which current is passed be $t=20s$
Let the total charge that passes through the battery in this time be $q$.
Therefore, using (1), we get
$7.2\times {{10}^{4}}=12\times I\times 20=240I$
$\therefore I=\dfrac{7.2\times {{10}^{4}}}{240}=300A$ --(3)
Now, using (2), we get
$q=It$
Putting (3) in the above equation, we get
$q=300\times 20=6000C$
Therefore, the total charge that passes through the battery to charge it is $6000C$.

So, the correct answer is “Option D”.

Note: This problem could also have been solved by realizing that the energy that is supplied to the battery can be written as the voltage of the battery and the total charge passing through it. This would enable us to do away with the current and time factors in the equations and give us our answer directly. This in fact would have been a shorter process and make our calculations much simpler. This result can be obtained by putting (2) in (1).