Electromagnetic Induction: Faraday and Lenz's Laws, Study notes of Physics

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CP2 ELECTROMAGNETISM

https://users.physics.ox.ac.uk/∼harnew/lectures/

LECTURE 15:

ELECTROMAGNETIC

INDUCTION

Neville Harnew^1

University of Oxford

HT 2022

1 1 With thanks to Prof Laura Herz

OUTLINE : 15. ELECTROMAGNETIC INDUCTION

15.1 Summarizing where we are

15.1.1 Electrostatics 15.1.2 Magnetostatics

15.2 Electromagnetic induction - outline

15.3 Faraday and Lenz’s Laws of Induction

15.3.1 Electromotive force (EMF) 15.3.2 Magnetic flux

15.4 Faraday’s and Lenz’s Laws

15.5 Faraday’s Law in differential form

15.1.2 Summarizing where we are : magnetostatics

1. Biot-Savart Law :

B(r) = 4 μ 0 π

ν

J(R) (r−R)^3 ×^ (r^ −^ R)^ dν I (^) There are no magnetic monopoles. Magnetic field lines form closed loops.

2. Gauss Law of magnetostatics : ∮

S

B · da = 0 ︸ ︷︷ ︸ integral form

→ ∇ · B = 0

differential form

3. Ampere’s Law : I (^) Magnetic fields are generated by electric currents. →

B · d` = μ 0 Iencl. → ∇ × B = μ 0 J

4. Continuity equation : I

S J^ ·^ da^ =^ −^

d dt

ν ρ(ν)^ dν^ →^ ∇ ·^ J^ =^ −^

d dt (ρ) (charge conserved)

Vector and scalar potential

Off syllabus, but worth a mention

Origins of electromagnetic induction

15.3 Faraday and Lenz’s Laws of Induction

15.3.1 Electromotive force (EMF)

I (^) Consider a wire moving with velocity v

through a B-field.

I (^) Free charges in the wire experience a

Lorenz force, perpendicular to v & B: F = q v × B

I (^) This moves charge to one side/end of the wire, which will create an electric potential drop along the wire : E =

`

dW q =^

`

F · d` q (by definition,^ V^ =^ work/unit charge ) I (^) Hence E =

(v^ ×^ B)^ ·^ d E is the electromotive force (or electromotance) (EMF) I (^) Note that E is not a force but a line integral over a force (i.e. a potential)!

15.4 Faraday’s and Lenz’s Laws

I Faraday’s Law

The induced electromotance (EMF) E in any closed

circuit is equal to (the negative of) the time rate of

change of the magnetic flux Φ through the circuit.

dΦ

dt =^

d dt

S B^ ·^ da^ =^ −E

I Lenz’s Law

The induced electromotance always gives rise to a

current whose magnetic field opposes the original

change in magnetic flux.

15.5 Faraday’s Law in differential form

I (^) Net potential around a closed circuit loop = 0

E =

E^ ·^ d,^ hence^ V^ =^ −E^ =^ −^

E^ ·^ d I (^) Faraday’s Law in integral form

E =

E^ ·^ d^ =^ −^

d dt

S B^ ·^ da Apply Stokes’ theorem to LHS : ∫ S (∇ ×^ E)^ ·^ da^ =^ −^

d dt

S B^ ·^ da I (^) Gives Faraday’s Law in differential form

∇ × E = −∂ ∂Bt

I (^) Any time-varying magnetic field (or change in magnetic

flux) generates an electric field which results in an electric potential E. (In contrast ∇ × E = 0 for electro/magnito-statics)