An external resistance R is connected to a battery of e.m.f. V and internal resistance r . The joule heat produced in resistor R is maximum when R is equal to

Options

A. r
B. `(r )/(2)`
C. 2r
D. Infinitely large

Answer

To solve the problem of finding the value of external resistance \( R \) that maximizes the Joule heat produced in it when connected to a battery with an electromotive force (e.m.f.) \( V \) and an internal resistance \( r \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Circuit**:
We have a battery with e.m.f. \( V \) and internal resistance \( r \) connected to an external resistor \( R \).

2. **Determine the Current**:
The total resistance in the circuit is the sum of the internal resistance and the external resistance, which is \( R + r \). The current \( I \) flowing through the circuit can be expressed using Ohm's law:
\[ I = \frac{V}{R + r} \]

3. **Calculate the Power (Joule Heat)**:
The power \( P \) dissipated in the external resistor \( R \) (which is the Joule heat produced) is given by:
\[ P = I^2 R \]
Substituting the expression for \( I \):
\[ P = \left(\frac{V}{R + r}\right)^2 R = \frac{V^2 R}{(R + r)^2} \]

4. **Maximize the Power**:
To find the value of \( R \) that maximizes \( P \), we can take the derivative of \( P \) with respect to \( R \) and set it to zero:
\[ \frac{dP}{dR} = 0 \]

5. **Differentiate the Power Expression**:
Using the quotient rule for differentiation:
\[ \frac{dP}{dR} = \frac{(R + r)^2 \cdot V^2 - V^2 R \cdot 2(R + r)}{(R + r)^4} \]
Simplifying the numerator:
\[ V^2 \left((R + r)^2 - 2R(R + r)\right) = V^2 \left(R^2 + 2Rr + r^2 - 2R^2 - 2Rr\right) = V^2 (r^2 - R^2) \]

6. **Set the Derivative to Zero**:
Setting the numerator equal to zero for maximization:
\[ r^2 - R^2 = 0 \]
This implies:
\[ r^2 = R^2 \implies R = r \]

7. **Conclusion**:
The Joule heat produced in the resistor \( R \) is maximum when the external resistance \( R \) is equal to the internal resistance \( r \).

### Final Answer:
The Joule heat produced in resistor \( R \) is maximum when \( R = r \).