An external resistance R is connected to a cell of internal resistance r. The current in the circuit is maximum when

Options

A. `R lt r`
B. `R gt r`
C. `R=r`
D. `R=2r`

Explanation

To determine when the current in a circuit is at its maximum when an external resistance \( R \) is connected to a cell with internal resistance \( r \), we can follow these steps:

### Step 1: Understand the Circuit
We have a cell with an electromotive force (emf) \( V \) and an internal resistance \( r \). An external resistance \( R \) is connected across the cell. The total resistance in the circuit is \( R + r \).

### Step 2: Write the Expression for Current
Using Ohm's Law, the current \( I \) flowing through the circuit can be expressed as:
\[ I = \frac{V}{R + r} \]

### Step 3: Write the Expression for Power
The power \( P \) delivered to the external resistance \( R \) can be expressed as:
\[ P = I^2 R \]
Substituting the expression for \( I \):
\[ P = \left(\frac{V}{R + r}\right)^2 R \]
This simplifies to:
\[ P = \frac{V^2 R}{(R + r)^2} \]

### Step 4: Differentiate the Power with Respect to \( R \)
To find the maximum power, we differentiate \( P \) with respect to \( R \) and set the derivative equal to zero:
\[ \frac{dP}{dR} = 0 \]

Using the quotient rule for differentiation:
\[ \frac{dP}{dR} = \frac{(R + r)^2 \cdot V^2 - V^2 R \cdot 2(R + r)}{(R + r)^4} \]

### Step 5: Set the Derivative to Zero
Setting the numerator equal to zero gives:
\[ (R + r)^2 - 2R(R + r) = 0 \]
Expanding and simplifying:
\[ R^2 + 2Rr + r^2 - 2R^2 - 2Rr = 0 \]
This simplifies to:
\[ -r^2 + R^2 = 0 \]
Thus, we have:
\[ R^2 = r^2 \]

### Step 6: Solve for \( R \)
Taking the square root of both sides, we find:
\[ R = r \]

### Conclusion
The current in the circuit is maximum when the external resistance \( R \) is equal to the internal resistance \( r \).