Question

In a circuit there is a battery with internal resistance \( r \) and Emf \( E \), which is connected to external load resistance \( R \) as shown. Find the value of \( R \) so that maximum power dissipates across \( R \).

• A. \( R = r \)
• B. \( R = \frac{r}{2} \)
• C. \( R = \sqrt{2}r \)
• D. \( R = 2r \)

Hint:

For maximum power dissipation, the load resistance should always equal the internal resistance of the source.

Answer 1

The Correct Option is A

To find the value of \( R \) for which maximum power dissipates across it, we use the concept of maximum power transfer theorem. According to this theorem, maximum power is transferred from the source to the load when the load resistance \( R \) is equal to the internal resistance \( r \) of the source. Below is the step-by-step explanation:

1. The total load in the circuit combines the internal resistance \( r \) and the external load resistance \( R \). Hence, the total resistance in the circuit is \( R_{\text{total}} = R + r \).
2. The current \( I \) flowing through the circuit can be determined using Ohm's Law: \(I = \frac{E}{R + r}\), where \( E \) is the electromotive force (EMF) of the battery.
3. The power \( P \) dissipated across the load resistance \( R \) is given by: \(P = I^2 R = \left(\frac{E}{R + r}\right)^2 R\).
4. To find the condition for maximum power, we differentiate the power \( P \) with respect to \( R \) and set this derivative to zero: \(\frac{dP}{dR} = 0\).
5. When you solve this derivative equation, it results in: \(R = r\).

Thus, the value of \( R \) for which the maximum power is dissipated across it is \( R = r \).