A DC source of EMF E volts and internal resistance R ohms is c... | General Science MCQ | Examoogle

General SciencePhysics: Electricity

A DC source of EMF E volts and internal resistance R ohms is connected to a variable load and it is adjusted such that the load absorbs maximum power from the source. The maximum power delivered from the source to the load is:

Option 1

Option 2

Option 3

Option 4

Correct Answer

⚙️ TE • Technical Concept & PrincipleGeneral SciencePhysics: Electricity

Option A

Option 1

Quick Summary:

According to the Maximum Power Transfer Theorem, a DC source delivers maximum power to a load resistance when the load resistance ( $R_L$ ) is exactly equal to the internal resistance of the source ( $R$ ). By substituting $R_L = R$ into the power formula, we derive the maximum power value.

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🔬SCScience SolutionConcept & Principle

💡 Explanation

💡 Everyday Analogy (Real-World Intuition)

Think of it like impedance matching in audio systems: for the best sound quality (maximum signal transfer), the output impedance of the amplifier must match the impedance of the speaker.

🧠 Memory Mnemonic / Shortcut Aid

Maximum Power requires 'Load matches Internal' ( $R_L = R$ )

🔢 Key Formulas

$P = I^2 R_L$ — Power dissipated by the load

$I = \frac{E}{R + R_L}$ — Current in the circuit

$P_{max} = \frac{E^2}{4 R}$ — Maximum power transfer formula

⚙️ Working Principle

The power delivered to a load is given by $P = I^2 R_L$ , where $I = \frac{E}{R + R_L}$ . Substituting $I$ , we get $P = \left(\frac{E}{R + R_L}\right)^2 R_L$ . To find the maximum, we set the derivative $\frac{dP}{dR_L} = 0$ , which yields $R_L = R$ . At this condition, the power becomes $P_{max} = \frac{E^2}{(R + R)^2} \times R = \frac{E^2 R}{4 R^2} = \frac{E^2}{4 R}$ .

📌 Key Points

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Maximum power transfer occurs when $R_L = R$
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At maximum power transfer, the efficiency is only 50%, as half the power is dissipated as heat within the internal resistance
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This theorem applies to both DC and AC circuits (where $Z_L = Z_{source}^*$ )

✅ Advantages

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Provides the theoretical limit for power extraction from a source with fixed internal resistance

❌ Disadvantages / Limitations

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Efficiency is low (50%) at maximum power transfer, which is undesirable for power transmission systems where energy conservation is prioritized over power maximization

🛠️ Applications / Uses

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Audio amplifier design for speaker matching
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RF antenna impedance matching
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Communication systems signal optimization

📄 Additional Information

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The internal resistance $R$ acts as a series constraint limiting the power output.
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Option A ( $\frac{E^2}{R}$ ) represents power when $R_L = 0$ (short circuit) which results in zero load power, Option B ( $\frac{E^2}{2R}$ ) is incorrect for load power, and Option D ( $\frac{2E^2}{R}$ ) is a dimensionally incorrect expression.

📊 Diagram / Illustration

✅

C is correct — The maximum power transferred from a DC source with EMF $E$ and internal resistance $R$ to a load is $\frac{E^2}{4R}$ .

Core Concepts Used

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Maximum Power Transfer Theorem Internal Resistance DC Circuit Analysis

💡 EXAM TIP

Always remember the difference between 'maximum power transfer' (condition $R_L = R$ ) and 'maximum efficiency' (condition $R_L \to \infty$ for near 100% efficiency).

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