Chapter 31 Faraday’s Law

1. Chapter 31Faraday’s Law • Faraday’s Law of Induction • Motional emf • Lenz’s Law • Induced emf and electric fields * This chapter explores the effects produced by magnetic fields that vary in time. * Michael Faraday (1831) showed that an emf can be induced in a circuit by a changing magnetic field.

2. 31.1 Induced EMF when a magnet is moved near a wire loop of area A, current flows through that wire without any batteries! ; moving a magnet means changing the B-field through the loop area  the magnetic flux changing Variable ΦB (increasing) Constant ΦB Variable ΦB (decreasing)

3. Induced EMF The needle deflects momentarily when the switch is closed; initially (the instant of closing switch), the magnetic field increases during a certain very small period of time (the needle deflects) until B-field becomes steady (the needle goes back to zero)

4. 31.2 Faraday’s Law of Induction The emf induced in a circuit is directly proportional to the time rate of change of the magnetic flux through the circuit. where, For N loops,

5. Faraday’s Law of Induction • To induce an emf we can change over time if the following: • the magnitude of B • the area enclosed by the loop • the angle between B and the normal to the area • any combination of the above

6. Ex: 31.1: a Way to Induce an emf in a Coil B-field ┴ to the plane. B-field changes linearly from 0 to 0.50 T in 0.80 s. What is the magnitude of the induced emf in the coil while the field is changing? square coil N=200 turns 18 cm The induced emf is B-field A=(0.18 m)² = 0.0324 m2 At t = 0 s, ΦB = 0 (no magnetic field) At t = 0.8 s, ΦB= B.A = BAcos0 = BA= (0.50 T)(0.0324 m²) = 0.0162 T.m²

7. I B Ex: 31.1:Exponentially Decaying Magnetic Field A wire loop of area A in a magnetic field B B┴ A Find the induced emf in the loop as a function of time.

8. Electrons moves through the conductor because of FB 31.2.Motional EMF * When B changes and the conductor is stationary  induced emf (examples befor) *When a conductor moves through a constant magnetic field motional emf,  charges moves in the direction of FB and leaves positive charges behind. As the wire moves,

9. 31.2.Motional EMF * When B changes and the conductor is stationary  induced emf (examples befor) *When a conductor moves through a constant magnetic field motional emf,  charges moves in the direction of FB and leaves positive charges behind. As the wire moves, As they accumulate on the bottom, an electric field is set up inside. In equilibrium, Voltage drop across the conductor a potential difference is maintained between the ends of the conductor as long as the conductor continues to move through the uniform magnetic field.

10. Motional EMF in a Circuit If the moving conductor is part of a closed conducting path (closed circuit of resistance R). The area enclosed by the circuit is A = lx If the bar is moved with constant velocity,

11. Magnetic Force on a Sliding Bar conducting bar moves on two frictionless parallel rails in a uniform magneticfield directed into the page. Using Newton’s laws, find the velocity of the bar as a function of time. , but The bar has a mass, m, and an initial velocity vi where - Sign indicate the force is to left

12. 31.3. Lenz’s Law The polarity of the induced emf is such that it tends to produce a current that creates a magnetic flux to oppose(يعاكس)the change in magnetic flux through the area enclosed by the current loop. As the bar is slid to the right, the flux through the loop increases. This induces an emf that will result in an opposing flux. Since the external field is into the screen, the induced field has to be out of the screen. Which means a counterclockwise current

13. Energy Considerations Suppose, instead of flowing counterclockwise, the induced current flows clockwise: Then the force will be towards the right which will accelerate the bar to the right which will increase the magnetic flux which will cause more induced current to flow which will increase the force on the bar … and so on the system wouldacquire energy with no input of energy. All this is inconsistent with the conservation of energy

14. Moving Magnet and Stationary Coil • Left moving magnet decreases flux through the loop. • It induces a current that creates it own magnetic field to oppose the flux decrease. • Right moving magnet increases flux through the loop. • It induces a current that creates it own magnetic field to oppose the flux increase.

15. Ex: 31.6:Application of Lenz’s Law When the switch is closed, the flux goes from zero to a finite value in the direction shown. To counteract this flux, the induced current in the ring has to create a field in the opposite direction. After a few seconds, since there is no change in the flux, no current flows. When the switch is opened again, this time flux decreases, so a current in the opposite direction will be induced to counter act this decrease.

16. Ex: 31.6: Loop Moving Through a Magnetic Field Find (A) the magnetic flux through the area enclosed by the loop, (B) the induced motional emf, and (C) the external applied force necessary to counter the magnetic force and keep v constant. Read carefully the text of the example from the book

17. 31.4.Induced EMF and Electric Fields Induced Electric Field is created Inside a Conductor Changing Magnetic Flux EMF This induced electric field is non-conservative and time-varying For any closed path, we find E by Hence, If the B changes with time Induced E-field General Form of Faraday’s Law Work for moving a charge q one cycle E is not conservative, because if it is conservative we will have A conducting loop of radius r with B changing with time For a circular loop

18. Ex: 31.8 Electric Field Induced by a Changing Magnetic Field in a Solenoid A long solenoid of radius R has n turns of wire per unit length and time varying current r>R Find E outside the solenoid at a distance r > R and inside the solenoid at r < R r<R

19. Summary *Faraday’s law of induction  induced emf *conducting bar of length l moves at a velocity v through a magnetic field B  motional emf induced in the bar *The applied force to keep constant v is Where R is resistance connected to moving bar *Lenz’s law states that the induced current and induced emf in a conductor are in such a direction as to set up a magnetic field that opposes the change in the magnetic flux *A general form of Faraday’s law of induction is Which implies induced E-field is not conservative

20. Discussion (1, 10, 15, 20, 33)

21. But, = First we need to find the change in the ΦB of the solenoid Which is the same flux that change in the coil

22. A

24. A at P1 E