LECTURE 16 General Review • Electrostatics • Faraday’s Law of Induction • motion of “q” in external E-field

• E-field generated by qi dA dΦ • Magnetostatics ε = − M • motion of “q” and “I” in external B-field B B dt • B-field generated by “I”

• Electrodynamics   • time dependent B-field generates E-field ΦM ≡ B • dA – ac circuits, inductors, transformers, etc ∫ – time dependent E-field generates B-field

Induction: DEMONSTRATION Induction Effects *Current flows only if there is Bar magnet moves through S N relative motion between the coil: Current induced in coil v loop and the magnet Change pole that enters: *Current disappears when the Induced current changes sign N S v relative motion ceases Bar magnet stationary inside *Faster motion produces a greater coil: No current induced in coil N S current Coil moves past fixed bar v magnet: Current induced in coil S N Caution – this picture is not an example of right hand rule! Change direction of motion in any of above examples  10/19/15 ammeter 3 10/19/15Induced current changes sign 4

1 Induction Effects from Currents Induction Example A wire loop falling into a B field (increasing)

Force acting on time moving charges magnetic downward www.daviddarling.info/images/electromagnetic_induction Field B Velocity v • When the switch is closed (or opened) current induced in coil b • Steady state current in coil a no current induced in coil b • Conclusion: A current is induced in a loop when: there is a change in magnetic field through it. N N N This can happen many different ways. • How can we quantify this? 5 10/19/15 6

Faraday’s Law Faraday’s Law Restated An emf is induced in a loop when the number of The magnitude of the emf  induced in a conducting magnetic field lines that pass through the loop is loop is equal to the rate at which the magnetic flux changing. M through the loop changes. dA Define the flux of the magnetic dA field through an open surface A ΔΦ as: B B ε = − B B B Δt ΦB = BAcosθ

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2 EMF for a Coil of N turns How to Change Magnetic Flux in a Coil

 ΔΦ ΔB 1. B changes: B = N Acosθ Δt Δt ΔΦ ΔA 2. A changes: B = NB cosθ Δt Δt ΔΦ Δ cosθ 3. θ changes: B = NBA [ ] Δt Δt ΔΦ ΔN 4. N changes (unlikely): B = BAcosθ 10/19/15 9 10/19/15 Δt Δt 10

Lenz’ Law Demo: E&M Cannon v An induced current has a direction such that the • Connect solenoid to a source of magnetic field due to the induced current opposes the alternating voltage. change in the magnetic flux that induces the current. • The flux through the area  to B ~ Opposition to Flux: axis of solenoid therefore changes in time side view • A conducting ring placed on top B of the solenoid will have a current N S induced in it opposing this v change. • There will then be a force on the ring since it contains a current which is circulating in the B presence of a magnetic field. N S v • Note that it’s the off-axis component of B (the “fringe field”) that flings the ring. 10/19/15 11 10/19/15 12

3 Faraday’s Law in terms of E Field Summary

x x xE x x x x x x x • Faraday’s Law (Lenz’ Law) E x x x x x x x x x x – a changing magnetic flux through a loop r x x x x x x x x x x induces a current in that loop B x x x x x x x x x x dΦ negative sign indicates that E ε = − M the induced EMF opposes x x x x x x x xE x x dt the change in flux • Faraday’s Law in terms of Electric Fields   dΦ ∫ E • dl = − M dt

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