CP2 ELECTROMAGNETISMhttps://users.physics.ox.ac.uk/∼harnew/lectures/

LECTURE 14:

SUMMARY OF MAGNETISM& INTRO TO INDUCTION

Neville Harnew1

University of Oxford

HT 2020

1With thanks to Prof Laura Herz1

OUTLINE : 14. SUMMARY OF MAGNETISM & INTROTO INDUCTION

14.1 Conservation of charge

14.2 Current density and Ohm’s Law

14.3 Vector and scalar potential

14.4 Let’s take stock of where we are : electrostatics

14.5 Let’s take stock of where we are : magnetostatics

14.6 Electromagnetic induction - outline

14.7 Origins of electromagnetic induction

14.8 Electromotive force (or electromotance)

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14.1 Conservation of charge

I Consider a volume ν bounded by asurface S.

I The integral of current density flowingout (or into) the surface J · da is equalto the charge lost by the volume [perunit time].

I∫

S J · da = I = −dQdt = − d

dt

∫ν ρ(ν)dν

Statement of the conservation of chargeI Use the divergence theorem on the

LHS∫ν ∇ · Jdν = − d

dt

∫ν ρ(ν)dν

This gives the continuity equation → ∇ · J = − ddt (ρ)

(mathematical statement of charge conservation)

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Summary : charge conservation & the continuity equation

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14.2 Current density and Ohm’s Law

I Ohm’s Law V = I R

V = E `

I = J A

→ E ` = J A RI This gives Ohm’s Law in terms of

current density: → J = `R A E

I Conductivity σ = `R A

Resistivity ρ = 1/σ

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14.3 Vector and scalar potentialOff syllabus, but worth a mention

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14.4 Let’s take stock of where we are : electrostatics1. Coulomb’s Law :

E(r) = 14π ε0

∫ν

ρ(R)(r−R)3 (r−R)dν

I An electric charge generates an electricfield. Electric field lines begin and end oncharge or at∞.

2. Gauss Law :∮SE · da = Qencl./ε0︸ ︷︷ ︸

integral form

→ ∇ ·E = ρ/ε0︸ ︷︷ ︸differential form

3. The electric field is conservative :I A well-defined potential V such that E = −∇V→

∮E · d` = 0 (work done is independent of path)

I Using the vector identity : ∇×E = −∇×∇V = 0I Hence ∇×E = 0

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14.5 Let’s take stock of where we are : magnetostatics1. Biot-Savart Law :

B(r) = µ04π

∫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 :∮SB · da = 0︸ ︷︷ ︸

integral form

→ ∇ ·B = 0︸ ︷︷ ︸differential form

3. Ampere’s Law :I Magnetic fields are generated by electric currents.

→∮B · d` = µ0 Iencl. → ∇×B = µ0 J

4. Continuity equation :I∫

S J · da = − ddt

∫ν ρ(ν)dν → ∇ · J = − d

dt (ρ)

(charge conserved)8

14.6 Electromagnetic induction - outline

Up to now we have considered stationary charges and steadycurrents. We now focus on what happens when either theE-field or B-field varies with time.

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14.7 Origins of electromagnetic induction

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14.8 Electromotive force (or electromotance)

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