Statement I: When an external resistor of resistance R (connected across a cell to internal resistance r ) is varied, power consumed by resistance R is maximum when R = r.
Statement II: Power consumed by a resistor of constant resistance R is maximum when current through it is maximum.

Options

A. Statement I is true, Statement II is True, Statement II is a correct explanation for Statement I.
B. Statement I is True, Statement II is True, Statement II is not a correct explanation for Statement I.
C. Statement I is True, Statement II is False.
D. Statement I is False, Statement II is True.

Explanation

To solve the problem, we need to analyze both statements and determine their validity based on the principles of electrical circuits.

### Step-by-Step Solution:

**Step 1: Analyze Statement I**
- Statement I claims that the power consumed by an external resistor \( R \) connected across a cell with internal resistance \( r \) is maximum when \( R = r \).
- The total current \( I \) flowing through the circuit can be expressed as:
\[ I = \frac{E}{R + r} \]
where \( E \) is the electromotive force (emf) of the cell.

**Step 2: Calculate Power across the External Resistor**
- The power \( P \) consumed by the external resistor \( R \) can be given by:
\[ P = I^2 R = \left(\frac{E}{R + r}\right)^2 R \]
Simplifying this, we get:
\[ P = \frac{E^2 R}{(R + r)^2} \]

**Step 3: Find Maximum Power Condition**
- To find the condition for maximum power, we differentiate \( P \) with respect to \( R \) and set the derivative to zero:
\[ \frac{dP}{dR} = 0 \]
- Using the quotient rule for differentiation, we can differentiate \( P \):
\[ \frac{dP}{dR} = \frac{(R + r)^2 \cdot E^2 - E^2 R \cdot 2(R + r)}{(R + r)^4} \]
Setting the numerator to zero gives:
\[ (R + r)^2 - 2R(R + r) = 0 \]
Simplifying this leads to:
\[ R + r - 2R = 0 \implies r = R \]
- This shows that the power is maximum when \( R = r \). Thus, Statement I is correct.

**Step 4: Analyze Statement II**
- Statement II states that the power consumed by a resistor of constant resistance \( R \) is maximum when the current through it is maximum.
- The power \( P \) across a resistor is given by:
\[ P = I^2 R \]
- Since \( R \) is constant, \( P \) is directly proportional to \( I^2 \). Therefore, if the current \( I \) is maximum, the power \( P \) will also be maximum. Hence, Statement II is also correct.

**Step 5: Conclusion**
- Both statements are correct, but Statement II does not explain Statement I. Therefore, the answer is that Statement I is correct and Statement II is also correct but does not explain Statement I.

### Final Answer:
- Statement I is correct.
- Statement II is correct but does not explain Statement I.

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