---
title: "Faraday's laws; mechanical to heat energy"
subject: "Engineering Physics"
chapter: 8
key: "semester_1/engineering_physics:faraday-law-mechanical-to-heat"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/faraday-law-mechanical-to-heat
site: BCT Engineering
---

# Faraday's laws; mechanical to heat energy

Engineering Physics · Chapter 8: Electromagnetism

- Asked in: 2068 Shrawan, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/faraday-law-mechanical-to-heat
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-8

## Question

State and explain Faraday laws of induction. Show that in induction, the mechanical energy is converted into electrical and finally into heat energy.

## Short answer

First law: a changing magnetic flux induces an emf. Second law: induced emf magnitude equals rate of change of flux linkage, ε = −N dΦ/dt. For a rod sliding on rails in uniform B, pulling work equals electrical power (BLv)^2/R which equals Joule heat I^2R, demonstrating conversion of mechanical energy to electrical and finally to heat.

## Answer

#### Faraday's laws of induction
- **First law:** whenever the magnetic flux linked with a circuit changes, an emf is induced in the circuit. The emf lasts only while the flux changes.
- **Second law:** the magnitude of the induced emf equals the rate of change of flux linkage: ε = −N dΦ_B/dt. The negative sign is Lenz's law: the induced current opposes the change that produces it.

Flux changes can occur by moving a magnet, changing B, changing the area, or changing the orientation of the loop.

#### Mechanical energy to heat
Take a conducting rod of length L pulled with constant speed v on rails in field B (into the page), the circuit having total resistance R.

```text
   x x x x x x x x
   x x x x x x x x
   ==+---------+====  rail
     |         |->  v (pulled by F_ext)
   R |         |
   ==+---------+====  rail
   x x x x x x x x   B into page
```

1. Flux = B L x, so ε = −dΦ/dt = −B L v. Magnitude: ε = BLv.
2. Current: I = ε/R = BLv/R.
3. Magnetic force on the rod: F = I L B = B²L²v/R. By Lenz's law it opposes the motion.
4. To keep v constant, an external agent must pull with F_ext = B²L²v/R.
5. Mechanical power supplied: P = F_ext v = B²L²v²/R.
6. Electrical power in the circuit: ε I = BLv · BLv/R = B²L²v²/R. Same as step 5.
7. Heat power: I²R = (BLv/R)² R = B²L²v²/R. Same again.

So P_mechanical = P_electrical = I²R: the work done by the agent becomes electrical energy and then Joule heat. Energy is conserved.

## Check yourself

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

1. Faraday's second law states that induced emf is:
   A. proportional to the flux itself
   B. equal to the rate of change of flux linkage, ε = −N dΦ/dt
   C. independent of the rate of change
   D. equal to the flux times the resistance
   Answer: B. The negative sign is Lenz's law. (Key point: Second law)

2. A rod of length L moving at speed v in field B has induced emf:
   A. BLv
   B. BL/v
   C. B²Lv
   D. BLv²
   Answer: A. Flux = BLx, so dΦ/dt = BLv. (Key point: Rod emf)

3. The external force needed to keep v constant is:
   A. BLv/R²
   B. zero
   C. B²L²v/R, balancing the magnetic force
   D. B L v R
   Answer: C. I = BLv/R and F = ILB. (Key point: Force needed)

4. What is shown about the powers?
   A. Mechanical power is greater than the heat
   B. Heat power is greater than the mechanical power
   C. They are unrelated
   D. Mechanical power = electrical power = I²R = B²L²v²/R
   Answer: D. All work done becomes Joule heat; energy is conserved. (Key point: Energy balance)

## Related in other subjects

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- [Soft magnetic material, Faraday's laws](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Asoft-magnetic-faraday-laws) — Electrical Machines, chapter 1
- [Faraday laws; static vs dynamic emf](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Afaraday-static-dynamic-emf) — Electrical Machines, chapter 1