The maximum power transfer occur in a cell when the external resistance is

1. Analyze the Question

The question asks for the condition under which a cell delivers the maximum amount of power to an external load (resistor). This is a classic application of the Maximum Power Transfer Theorem in electrical circuits.

2. Explain the Concept

In a simple circuit, a cell has an electromotive force ($E$) and an internal resistance ($r$). When you connect an external resistor ($R$) to this cell, the current ($I$) flowing through the circuit is given by Ohm's Law:

The power ($P$) dissipated by the external resistor $R$ is calculated using the formula:

Substituting the expression for $I$:

To find the value of $R$ that gives maximum power, we look for the point where the derivative of $P$ with respect to $R$ is zero ($\frac{dP}{dR} = 0$). Mathematically, this occurs when the external resistance $R$ is exactly equal to the internal resistance $r$.

3. Show the Solution

• Step 1: Identify the power formula: $P = \frac{E^2 R}{(R + r)^2}$
• Step 2: Understand that for maximum power, the load resistance must match the source resistance. If $R$ is too small, the total current is high, but the voltage drop across the load is low. If $R$ is too large, the voltage drop is high, but the current is very low.
• Step 3: The balance point where the product of current squared and resistance is maximized is when $R = r$.

4. Identify the Answer

Based on the derivation above, the maximum power transfer occurs when the external resistance is equal to the internal resistance of the cell.

The correct option is B.

5. Exam Tip

Think of this like a "matching" game. To get the most "work" (power) out of a battery, the resistance of the device you are powering must "match" the internal resistance of the battery itself. If you ever forget this, remember that Maximum Power = Matching Resistance.