Two cells , each of `emf E` and internal resistance `r`, are connected in parallel across a resistor `R`. The power delivered to the resistor is maximum if `R` is equal to

Options

A. R = r/2
B. R = r
C. R = 2r
D. R = 0

Answer

To solve the problem, we need to find the value of the resistor \( R \) that maximizes the power delivered to it when two identical cells, each with an electromotive force (emf) \( E \) and internal resistance \( r \), are connected in parallel.

### Step-by-Step Solution:

1. **Understand the Configuration**:
- We have two cells in parallel, each with emf \( E \) and internal resistance \( r \).
- The equivalent emf of the parallel combination remains \( E \).
- The equivalent internal resistance \( r_{eq} \) of two identical resistors in parallel is given by:
\[ r_{eq} = \frac{r}{2} \]

2. **Determine the Total Resistance**:
- The total resistance in the circuit when connected to the external resistor \( R \) is:
\[ R_{total} = r_{eq} + R = \frac{r}{2} + R \]

3. **Calculate the Current**:
- The current \( I \) flowing through the circuit can be calculated using Ohm's law:
\[ I = \frac{E}{R_{total}} = \frac{E}{\frac{r}{2} + R} \]

4. **Power Delivered to the Resistor**:
- The power \( P \) delivered to the resistor \( R \) is given by:
\[ P = I^2 R = \left(\frac{E}{\frac{r}{2} + R}\right)^2 R \]

5. **Maximize the Power**:
- To find the value of \( R \) that maximizes \( P \), we differentiate \( P \) with respect to \( R \) and set the derivative equal to zero:
\[ P = \frac{E^2 R}{\left(\frac{r}{2} + R\right)^2} \]
- Using the quotient rule for differentiation:
\[ \frac{dP}{dR} = \frac{\left(\frac{r}{2} + R\right)^2 \cdot E^2 - E^2 R \cdot 2\left(\frac{r}{2} + R\right)}{\left(\frac{r}{2} + R\right)^4} \]
- Setting the numerator equal to zero for maximization:
\[ \left(\frac{r}{2} + R\right)^2 - 2R\left(\frac{r}{2} + R\right) = 0 \]

6. **Simplifying the Equation**:
- This simplifies to:
\[ \frac{r}{2} + R - 2R = 0 \]
- Rearranging gives:
\[ \frac{r}{2} - R = 0 \implies R = \frac{r}{2} \]

### Final Answer:
The power delivered to the resistor is maximum when:
\[ R = \frac{r}{2} \]