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Faraday's laws; mechanical to 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.

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

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.

   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.

✓ Answer checked by Claude Opus 5.5 (high effort)

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