---
title: "Maximum power transfer theorem proof"
subject: "Basic Electrical Engineering"
chapter: 3
key: "semester_1/basic_electrical_engineering:max-power-transfer-theorem-prove"
source: https://www.bctengineer.com/q/semester_1/basic_electrical_engineering/max-power-transfer-theorem-prove
site: BCT Engineering
---

# Maximum power transfer theorem proof

Basic Electrical Engineering · Chapter 3: Network Theorems

- Asked in: 2072 Kartik, 6 marks; 2067 Asoj, 4 marks (asked 2 times)
- Video: [Maximum Power Transfer Theorem](https://www.youtube.com/watch?v=U85eA3-suiQ) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/basic_electrical_engineering/max-power-transfer-theorem-prove
- Whole chapter: https://www.bctengineer.com/md/semester_1/basic_electrical_engineering/chapter-3

## Question

State Maximum Power Transfer Theorem and also prove "maximum power will be dissipated when R_internal = R_L".

## Short answer

Maximum power transfer theorem states that maximum power is delivered to a load when its resistance equals the Thevenin resistance of the source network, i.e., R_L = R_th. By setting derivative of power P = E²R_L/(R_th+R_L)² to zero, we get R_L = R_th.

## Answer

#### Statement
**Maximum power transfer theorem:** a load receives maximum power from a source (of fixed EMF *E* and internal resistance *R_int*) when the load resistance equals the internal (Thevenin) resistance of the source: R_L = R_int.

```text
        R_int
 o--[ ]----+------o
   (E)     |
          [R_L]
 o---------+------o
```

#### Proof
Current: I = E/(R_int + R_L)

Power in the load: P = I² R_L = E² R_L/(R_int + R_L)²

For maximum power, dP/dR_L = 0:

dP/dR_L = E² [(R_int + R_L)² − 2 R_L (R_int + R_L)]/(R_int + R_L)⁴
dP/dR_L = E² (R_int − R_L)/(R_int + R_L)³

Setting this to zero gives R_int − R_L = 0, so **R_L = R_int**.

Check that it is a maximum: for R_L < R_int, dP/dR_L > 0 (power rises); for R_L > R_int, dP/dR_L < 0 (power falls). So the stationary point is a maximum.

#### Maximum power
P_max = E² R_int/(2 R_int)² = **E²/(4 R_int)**

#### Efficiency
Efficiency = P_load/P_total = R_L/(R_int + R_L) = 50 % at maximum power. So the theorem is used where the power is small and matching matters (communication circuits), not in power systems.

Example: E = 12 V, R_int = 3 Ω gives R_L = 3 Ω, I = 2 A, P_max = 12²/(4 × 3) = 12 W.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. When is maximum power transferred to the load?
   A. When R_L is much larger than R_int
   B. When R_L is zero
   C. When R_L equals the source internal resistance
   D. When R_L is half of R_int
   Answer: C. Maximum power occurs at R_L = R_int. (Key point: Statement)

2. What condition is used in the proof?
   A. P = 0
   B. d2P/dR_L2 = 0
   C. I = 0
   D. dP/dR_L = 0
   Answer: D. Set the derivative of load power to zero to find the stationary point. (Key point: Proof step)

3. What is P_max?
   A. E^2/(4 R_int)
   B. E^2/R_int
   C. E^2/(2 R_int)
   D. E^2/(8 R_int)
   Answer: A. Substituting R_L = R_int gives E^2 R/(2R)^2 = E^2/(4R). (Key point: Maximum power)

4. What is the efficiency at maximum power transfer?
   A. 100 %
   B. 50 %
   C. 25 %
   D. 75 %
   Answer: B. R_L/(R_int + R_L) = 1/2, so it is used where power is small, not in power systems. (Key point: Efficiency)

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