Question

A battery having emf $10\text{V}$ and internal resistance $2\Omega$ is connected to an external circuit as shown in the diagram. Find the value of $R$ for which maximum power will be transferred to the external circuit. Also find the efficiency of the battery.

• A. $3\Omega ,50$%
• B. $4\Omega ,50$%
• C. $2\Omega ,67$%
• D. $4\Omega ,67$%

Answer 1

The correct option is A $3\Omega ,50$%

We know that for maximum power trasfer, external resistance should be equal to internal resistance of the battery.

Consider the external circuit, we can observe that $3\Omega$ and $R$ are connected in series.

These resistances are in parallel combination. Their equivalent resistance, $R_{eq}=\frac{3\times (3+R)}{3+(3+R)}$.

From maximum power transfer theorem, $R_{eq}=2$

$\Rightarrow \frac{3\times (3+R)}{3+(3+R)}=2$

$\Rightarrow 9+3R=12+2R$

$\therefore R=3\Omega$

Current in the circuit, $i=\frac{10}{2+2}=2.5\text{A}$
Now, Efficiency $=\frac{P_{out}}{P_{in}}\times 100$

Input power, $P_{in}=E\times i=10\times 2.5=25\text{W}$
Output power, $P_{out}=i^2{R}_{eq}={2.5}^2\times 2=12.5\text{W}$
Therefore, efficiency $=50$%