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# Maxwell equation

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• ### Maxwell equation

1. 1. Induced FieldsMagnetic fields may vary in time.Experiments conducted in 1831 showed that an emf can be induced in acircuit by a changing magnetic field. Experiments were done by Michael Faraday and Joseph Henry.The results of these experiments led to Faraday’s Law of Induction.An induced current is produced by a changing magnetic field.There is an induced emf associated with the induced current.A current can be produced without a battery present in the circuit.Faraday’s law of induction describes the induced emf.
2. 2. Michael Faraday1791 – 1867British physicist and chemistGreat experimental scientistContributions to early electricity include: Invention of motor, generator, andtransformer Electromagnetic induction Laws of electrolysis
3. 3. EMF Produced by a Changing Magnetic Field, 1A loop of wire is connected to asensitive ammeter.When a magnet is moved toward theloop, the ammeter deflects. The direction was arbitrarilychosen to be negative.
4. 4. EMF Produced by a Changing Magnetic Field, 2When the magnet is held stationary,there is no deflection of the ammeter.Therefore, there is no induced current. Even though the magnet is in theloop
5. 5. EMF Produced by a Changing Magnetic Field, 3The magnet is moved away from theloop.The ammeter deflects in the oppositedirection.
6. 6. Induced Current Experiment, Summary
7. 7. EMF Produced by a Changing Magnetic Field, SummaryThe ammeter deflects when the magnet is moving toward or away from the loop.The ammeter also deflects when the loop is moved toward or away from themagnet.Therefore, the loop detects that the magnet is moving relative to it. We relate this detection to a change in the magnetic field. This is the induced current that is produced by an induced emf.
8. 8. Faraday’s Experiment – Set UpA primary coil is connected to a switchand a battery.The wire is wrapped around an ironring.A secondary coil is also wrappedaround the iron ring.There is no battery present in thesecondary coil.The secondary coil is not directlyconnected to the primary coil.
9. 9. Faraday’s ExperimentClose the switch and observe the current readings given by the ammeter.
10. 10. Faraday’s Experiment – FindingsAt the instant the switch is closed, the ammeter changes from zero in onedirection and then returns to zero.When the switch is opened, the ammeter changes in the opposite direction andthen returns to zero.The ammeter reads zero when there is a steady current or when there is nocurrent in the primary circuit.
11. 11. Faraday’s Experiment – ConclusionsAn electric current can be induced in a loop by a changing magnetic field. This would be the current in the secondary circuit of this experimental set-up.The induced current exists only while the magnetic field through the loop ischanging.This is generally expressed as:an induced emf is produced in the loop by thechanging magnetic field. The actual existence of the magnetic flux isnot sufficient to produce the induced emf, theflux must be changing.
12. 12. Faraday’s Law of Induction – StatementsThe emf induced in a circuit is directly proportional to the time rate of change ofthe magnetic flux through the circuit.Mathematically,Remember ΦB is the magnetic flux through the circuit and is found byIf the circuit consists of N loops, all of the same area, and if ΦB is the flux throughone loop, an emf is induced in every loop and Faraday’s law becomesBdεdtΦ= −Bdε NdtΦ= −dSB •=Ψ ∫dtddlEΨ−=•∫ ______________(1)
13. 13. Relation between EMF and Electrostatic field
14. 14. Faraday’s Law – ExampleAssume a loop enclosing an area A lies in auniform magnetic field.The magnetic flux through the loop is ΦB = BAcos θ.The induced emf isVemf = - d/dt (BA cos θ).Section 31.1
15. 15. Nature of a changing fluxNature of a changing fluxSince flux is defined as a dot product B can change A can change q can change∫ ⋅−=SdSBdtdNVemfdSB•=Ψ ∫ _______________(2)
16. 16. Polarity of the Induced EmfThe polarity (direction) of the induced emf isdetermined by Lenz’s law.
17. 17. LENZ’S LawThe direction of theThe direction of theemf induced byemf induced bychanging flux willchanging flux willproduce a currentproduce a currentthat generates athat generates amagnetic fieldmagnetic fieldopposing the fluxopposing the fluxchange thatchange thatproduced itproduced it..
18. 18. Lenz’s LawB, HLenz’s Law: emf appears and current flows that creates amagnetic field that opposes the change – in this case andecrease – hence the negative sign in Faraday’s Law.B, HN SV+, V-Iinduced
19. 19. Lenz’s LawB, HLenz’s Law: emf appears and current flows that creates amagnetic field that opposes the change – in this case anincrease – hence the negative sign in Faraday’s Law.B, HN SV-, V+Iinduced
20. 20. Faraday’s Law for a Single LoopdtdEΨ−== ε
21. 21. Faraday’s Law for a coil having NFaraday’s Law for a coil having NturnsturnsdtdNEΨ−==ε
22. 22. Ways of Inducing an emf1. A stationary loop in a time varying magnetic field B [TRANSFORMER EMF]i.e. The magnitude of the magnetic field can change with time.2. A time varying loop area in a static field B [ MOTIONAL EMF]i.e., The area enclosed by the loop can change with time.3. Time varying loop area in a time varying magnetic fieldi.e., The angle between the magnetic field and the normal to the loop can changewith time.Any combination of the above can occur.
23. 23. Motional emfA motional emf is the emf induced in aconductor moving through a constantmagnetic field.The electrons in the conductorexperience a force, that isdirected along ℓ .q= ×F v Br rr
24. 24. Motional emfMotional emfApply the Lorentz ForceApply the Lorentz Forcequation:quation:0qvBqEF =−=vBE = vBE =ε=vB=εqvBqE =
25. 25. Faraday’s Lawmd d dxBlx Bldt dt dtΦ= =dxBlv Bldt= =EmTherefore,ddtΦ=ECONCLUSION: to produce emf one should makeANY change in a magnetic flux with time!Consider the loop shown:
26. 26. Sliding Conducting BarA conducting bar moving through a uniform field and the equivalent circuitdiagram.Assume the bar has zero resistance.The stationary part of the circuit has a resistance R.
27. 27. Moving Conductor, VariationsUse the active figure to adjust theapplied force, the electric field and theresistance.Observe the effects on the motion ofthe bar.
28. 28. Sliding Conducting Bar, cont.The induced emf isSince the resistance in the circuit is R, the current isBd dxε B B vdt dtΦ= − = − = − Iε B vR R= =
29. 29. Maxwell’s Equations