1. Electromagnetic Induction and Faraday’s Law
Chapter 21
Electromagnetic Induction and Faraday’s Law

2. A bit of review of Ch 20 …

3. There is a counterclockwise current I in a circular loop of wire situated in an external magnetic field directed out of the page as shown above. The effect of the forces that act on this current is to make the loop?
expand in size
contract in size
rotate about an axis perpendicular to the page
rotate about an axis in the plane of the page
accelerate into the page

4. Which of the paths above represents the path of an electron traveling without any loss of energy through a uniform magnetic field directed into the page? A B C D E

5. Induced EMF
Almost 200 years ago, Faraday looked for evidence that a magnetic field would induce an electric current with this apparatus:

6. Induced EMF
He found no evidence when the current was steady, but did see a current induced when the switch was turned on or off.

7. Induced EMF
Induced EMF: the potential difference created by a changing magnetic field (flux) that causes a current to flow in a wire.
The induced EMF in a wire loop is proportional to the rate of change of magnetic flux (either the magnitude of the field or the Area or both)through the loop.
Magnetic flux:
(21-1)
Unit of magnetic flux: weber, Wb.
1 Wb = 1 T·m2

8. Faraday’s Law of Induction
This drawing shows the variables in the flux equation:

9. Magnetic Flux
The product of magnetic field and area.
Can be thought of as a total magnetic
“effect” on a coil of wire of a given area.

10. Maximum Flux
The area is aligned so that a
perpendicular to the area points parallel
to the field

11. Minimum Flux
The area is aligned so that a perpendicular to the area points perpendicular to the field

12. Intermediate Flux
The area is neither perpendicular nor is it parallel

13. Faraday’s Law of Induction
The magnetic flux is proportional to the total number of lines passing through the loop.

14. Magnetic Flux
ΦB = B A cosθ
– ΦB: magnetic flux in Webers (Tesla meters2)
– B: magnetic field in Tesla
– A: area in meters2.
– θ: the angle between the area and the
magnetic field.
ΦB = B·A

15. Sample Problem: Calculate the magnetic flux through a rectangular wire frame 3.0 m long and 2.0 m wide if the magnetic field through the frame is 4.2 mT.
a) Assume that the magnetic field is perpendicular to the area vector.
b) Assume that the magnetic field is parallel to the area vector.
c) Assume that the angle between the magnetic field and the area vector is 30o.

16. Sample Problem: Assume the angle is 40o, the magnetic field is 50 mT, and the flux is 250 mWb. What is the radius of the loop?

17. Induced Electric Potential
Armed with definition of flux, we can now look at the induced current due to a changing magnetic flux.
A system will respond so as to oppose changes in magnetic flux.
A change in magnetic flux will be partially offset by an induced magnetic field whenever possible.
Changing the magnetic flux through a wire loop causes current to flow in the loop.
This is because changing magnetic flux induces an electric potential.

18. Faraday’s Law of Induction
– Ɛ: induced potential (V)
– N: # loops
– ΦB: magnetic flux (Webers, Wb)
– t: time (s)
To generate voltage, change B, change A, change θ

19. An emf can be induced in one of three ways:
if the magnitude of the field changes
If the area of the loop changes:
If the angle between the loop face and the direction of the field changes

20. Magnetic flux will change if the angle between the loop and the field changes:

21. Sample Problem: A coil of radius 0
Sample Problem: A coil of radius 0.5 m consisting of 1000 loops is placed in a 500 mT magnetic field such that the flux is maximum. The field then drops to zero in 10 ms. What is the induced potential in the coil?

22. Sample Problem: A single coil of radius 0
Sample Problem: A single coil of radius 0.25 m is in a 100 mT magnetic field such that the flux is maximum. At time t = 1.0 seconds, field increases at a uniform rate so that at 11 seconds, it has a value of 600 mT. At time t = 11 seconds, the field stops increasing. What is the induced potential?
A) at t = 0.5 seconds?
B) at t = 3.0 seconds?
C) at t = 12 seconds?

23. Lenz’s Law
Experiments show that the newly induced current will flow in a direction so it’s created magnetic field will try to oppose the change in flux).
Use in combination with hand rule to predict current direction.

24. Problem Solving: Lenz’s Law
Point your right thumb in the initial direction of the magnetic field
Ask yourself, is the flux increasing or decreasing?
If the flux is decreasing, then just curl your fingers (with your thumb still pointed in the direction of the magnetic field). Your fingers show the direction of the induced current. (This is just the RHR that you’ve already learned.)
If flux is increasing in the direction you’re pointing, then flux is decreasing in the other direction. So point your thumb in the opposite direction of the magnetic field, and curl your fingers. Your fingers show the direction of the induced current.
Remember that the external field and the field due to the induced current are different.

25. Sample Problem: The magnetic field is increasing at a rate of 4.0 mT/s. What is the direction of the current in the wire loop?

26. Sample Problem: The magnetic field is increasing at a rate of 4.0 mT/s. What is the direction of the current in the wire loop?

27. Sample Problem: The magnetic field is decreasing at a rate of 4.0 mT/s. The radius of the loop is 3.0 m, and the resistance is 4 Ω. What is the magnitude and direction of the current?

28. Motional emf Ɛ = BLv – B: magnetic field (T)
– L: length of bar moving through field
– v: speed of bar moving through field.
Bar must be “cutting through” field lines. It cannot be moving parallel to the field.
This formula is easily derivable from Faraday’s Law of Induction

29. Motional emf – derivation
Ɛ = ΔΦB / Δt
Ɛ = Δ(BA) / Δt (assume cosθ = 1)
Ɛ = Δ(BLx) / Δt
Ɛ = BL Δx / Δt
Ɛ = BLv

30. 21.3 EMF Induced in a Moving Conductor
This image shows how the magnetic flux can change in a rectangular wire:
ε = BLv

31. 21.3 EMF Induced in a Moving Conductor
The induced current is in a direction that tends to slow the moving bar – it will take an external force to keep it moving.

34. Railgun