The maximum power dissipated by external resistance R by a cell of an external omf E and internal resistance r is `E^(2)//4R` which is obtained for

Options

A. `Rltr`
B. `Rltr`
C. `R=r`
D. any value of R.

Answer

To solve the problem of finding the condition for maximum power dissipated by an external resistance \( R \) in a circuit with a cell of EMF \( E \) and internal resistance \( r \), we can follow these steps:

### Step 1: Understand the Circuit
We have a circuit consisting of a cell with an EMF \( E \) and an internal resistance \( r \). The external resistance connected to this cell is \( R \).

### Step 2: Write the Expression for Current
The total resistance in the circuit is the sum of the internal and external resistances:
\[ R_{\text{total}} = R + r \]
The current \( I \) flowing through the circuit can be expressed using Ohm's law:
\[ I = \frac{E}{R + r} \]

### Step 3: Write the Expression for Power Dissipated in the External Resistance
The power \( P \) dissipated in the external resistance \( R \) can be calculated using the formula:
\[ P = I^2 R \]
Substituting the expression for \( I \):
\[ P = \left(\frac{E}{R + r}\right)^2 R \]
This simplifies to:
\[ P = \frac{E^2 R}{(R + r)^2} \]

### Step 4: Maximize the Power
To find the maximum power, we need to differentiate \( P \) with respect to \( R \) and set the derivative equal to zero:
\[ \frac{dP}{dR} = 0 \]
Using the quotient rule, we differentiate:
\[ \frac{dP}{dR} = \frac{(R + r)^2 \cdot E^2 - E^2 R \cdot 2(R + r)}{(R + r)^4} \]
Setting the numerator equal to zero:
\[ (R + r)^2 E^2 - 2E^2 R(R + r) = 0 \]
This simplifies to:
\[ E^2 \left( (R + r)^2 - 2R(R + r) \right) = 0 \]
Ignoring \( E^2 \) (since it cannot be zero), we have:
\[ (R + r)^2 - 2R(R + r) = 0 \]

### Step 5: Solve the Equation
Expanding and simplifying:
\[ R^2 + 2Rr + r^2 - 2R^2 - 2Rr = 0 \]
This reduces to:
\[ -r^2 + R^2 = 0 \]
Thus:
\[ R^2 = r^2 \implies R = r \]

### Conclusion
The maximum power is dissipated when the external resistance \( R \) is equal to the internal resistance \( r \).

### Final Answer
The maximum power dissipated by the external resistance \( R \) occurs when \( R = r \).

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