Walker, Chapter 23 Magnetic Flux and Faraday s Law of Induction

1. Walker, Chapter 23Magnetic Flux and Faraday�s Law of Induction

2. Magnetic Induction Demonstrations Ammeter for overhead projector which measures the current in a coil. Under what circumstances is a current induced in the coil? How do we get the largest current?

3.  Chapter 22: Electric currents (in a wire, in a plasma, in a fluid solution, inside an atom) produce a disturbance in the surrounding space called the magnetic field. This magnetic field produces forces on any other macroscopic or microscopic currents. Example: MRI: Magnetic field (several Tesla) from superconducting solenoid induces a net alignment of the microscopic currents inside each and every proton at the center of the Hydrogen atoms in your body

4. Induced emf (Voltage) from changing Magnetic Flux Electric currents produce magnetic fields. 19th century puzzle: Can magnetic fields produce currents? A static magnet will produce no current in a stationary coil. Faraday: If the magnetic field changes, or if the magnet and coil are in relative motion, there will be an induced emf (and therefore current) in the coil. Key Concept: The magnetic flux through the coil must change. This will induce an emf e in the coil, which produces a current I = e/R in the coil. Such a current is said to be induced by the varying B-field.

5. Magnetic Flux

6. Current Loop

7. Walker Problem 3, pg. 778

8. Faraday�s Law of Induction Faraday�s Law: The instantaneous emf in a circuit (w/ N loops) equals the rate of change of magnetic flux through the circuit:

9. Examples of Induced Current

10. Induced Current The current in the primary polarizes the material of the core. The magnetic field of the primary solenoid is enhanced by the magnetic field produced by these atomic currents. This magnetic field remains confined in the iron core, and only fans out and loops back at the end of the core. Any change in the current in the primary (opening or closing switch) produces a change in the magnetic flux through the secondary coil. This induces a current in the secondary.

11. Induction by Relative Motion When a permanent magnet moves relative to a coil, the magnetic flux through the coil changes, inducing an emf in the coil. In a) the magnitude of the flux is increasing In c) the flux is decreasing in magnitude. In a) and c) the induced current has opposite sign.

12. Induction by Rotational Motion As a coil rotates in a constant magnetic field (uniform or not) the flux through the loop changes, inducing an emf in the coil.

13. Walker Problem 10, pg. 778 This is a plot of the magnetic flux through a coil as a function of time. At what times shown in this plot does (a) the magnetic flux and (b) the induced emf have the greatest magnitude?

14. Walker Problem 9, pg. 778 A 0.25 T magnetic field is perpendicular to a circular loop of wire with 50 turns and a radius 15 cm. The magnetic field is reduced to zero in 0.12 s. What is the magnitude of the induced emf?

15. Lenz�s Law

16. Walker Problem 24, pg. 779

17. Motional emf An emf will also be produced if a conductor moves through a magnetic field. The emf comes from the motion of charges, which are free to move in the conductor. In this example, why does the top of the rod become positively charged?

18. If the moving conductor is part of a circuit, the flux through the circuit will change with time and a current will be induced (Area of loop = Ls):

19. Walker Problems 30-31, pg. 780

20. Eddy Currents

21. Generators A generator converts mechanical energy to electrical energy. Consider a current loop which rotates in a constant magnetic field: The magnetic flux through the loop changes, so an emf is induced. If a loop of area A with N turns rotates with angular speed w (period of rotation = 2p/w) in a constant B field, then the instantaneous induced emf is: = NBAw sin(wt) If this loop is part of a circuit, this emf will induce an Alternating Current (AC) in the circuit.

22. Generator

23. Self-Inductance If you try to change the current in a circuit instantaneously, the response will instead be gradual. This is because the circuit produces a self-induced emf to initially oppose any changes as prescribed by Lenz�s Law. This effect is known as self-induction. This does not violate the Newtonian principle of no-self-forces, because in effect individual electrons in the current are exerting forces on the other electrons in the same circuit.

24. Inductance The self induced emf is given by: where L is called the inductance of the circuit. The magnetic flux through the loop, produced by current in the loop, is proportional to the current. The inductance L is the constant of proportionality. The unit of inductance is the Henry 1 H = 1 T�m2/A = 1 (T�m2/s) (s/A) = 1 V�s/A Note that inductance, like capacitance, is purely geometrical.

25. Inductance of a Solenoid A solenoid has inductance given by

26. Walker Problem 41, pg. 780

27. RL Circuits We can construct a circuit from inductors and resistors. The circuit will behave just like an RC circuit, with a time constant given by: t = L/R

28. Walker Problem 45, pg. 780

29. Energy Stored in an Inductor Just as energy can be stored in a capacitor (recall that U= �CV2), energy can also be stored in an inductor: U = �LI2 Whereas energy in a capacitor is stored in the electric field between the plates, energy in an inductor is stored in the magnetic field within the inductor.

30. Transformers

31. Walker Problem 57, pg. 781