1. General Review Electrostatics Magnetostatics Electrodynamics
motion of “q” in external E-field
E-field generated by qi
Magnetostatics
motion of “q” and “I” in external B-field
B-field generated by “I”
Electrodynamics
time dependent B-field generates E-field
ac circuits, inductors, transformers, etc
time dependent E-field generates B-field

2. LECTURE 16
Faraday’s Law of Induction dA B

3. Induction: DEMONSTRATION
*Current flows only if there is relative motion between the loop and the magnet
*Current disappears when the relative motion ceases
*Faster motion produces a greater current
DEMOS: 6D-04 – Earth’s magnetic field – hoop and bar magnetic
Caution – this picture is not an example of right hand rule!
ammeter
11/7/2018

4. Induction Effects
Bar magnet moves through coil: Current induced in coil v S N
Change pole that enters: Induced current changes sign v N S
Bar magnet stationary inside coil: No current induced in coil
DEMO: 6D-04 Earth’s Magnet Field – hoop and bar magnet N S
Coil moves past fixed bar magnet: Current induced in coil v S N
Change direction of motion in any of above examples  Induced current changes sign
11/7/2018

5. Induction Effects from Currents
When the switch is closed (or opened) current induced in coil b
Steady state current in coil a no current induced in coil b
DEMO: 6D- 01 & 6D-02
Conclusion:
A current is induced in a loop when: there is a change in magnetic field through it.
This can happen many different ways.
How can we quantify this?

6. A wire loop falling into a B field (increasing)
Induction Example
A wire loop falling into a B field (increasing)
downward
Velocity v
magnetic
Field B
Force acting on
moving charges
time N N N
11/7/2018

7. Faraday’s Law
An emf is induced in a loop when the number of magnetic field lines that pass through the loop is changing.
Only components perpendicular to loop dA B
Define the flux of the magnetic field through an open surface A as:
ITEMS IN RED BOLD ARE FOR WRITING/DiSCUSSING IN CLASS. REMOVE FROM POSTED NOTES
Unit: 1 Weber=1Wb=1 Tm2
11/7/2018

8. Faraday’s Law Restated
The magnitude of the emf  induced in a conducting loop is equal to the rate at which the magnetic flux M through the loop changes. dA B
The minus sign indicates opposition
11/7/2018

9. EMF for a Coil of N turns
11/7/2018

10. How to Change Magnetic Flux in a Coil
DEMO: 6D-06 Search Coil
11/7/2018

11. Lenz’ Law (to determine direction of induced I in loop)
An induced current has a direction such that the magnetic field due to the induced current opposes the change in the magnetic flux that induces the current.
Opposition to Flux:
B induced always opposes the change in the flux of B, but does not always point opposite it!!! v B S N
*Move N towards loop. FluxB increases. Induced I sets up its own field to oppose the increase in FluxB. I must be counterclockwise. Bi
*Move N away from loop. FluxB decreases. Induced I sets up its own field to oppose the decrease in FluxB. I must be clockwise. v B S N
Direction of I must be to oppose change...otherwise violate conservation of energy.
11/7/2018

12. Lenz’ Law
An induced current has a direction such that the magnetic field due to the induced current opposes the change in the magnetic flux that induces the current.
Opposition to Flux: v B S N v B S N
11/7/2018

13. What is the direction of the induced current in the loop?
Example 1
X X X X X X X X X X X X v x y
A conducting rectangular loop moves with constant velocity v in the +x direction through a region of constant magnetic field B in the -z direction as shown.
What is the direction of the induced current in the loop?
No induced current. B is constant, A is constant, v is constant. dFluxB/Dt=0
(a) ccw
(b) cw
(c) no induced current
11/7/2018

14. Example 2
A conducting rectangular loop moves with constant velocity v in the -y direction and a constant current I flows in the +x direction as shown.
What is the direction of the induced current in the loop? v I x y
Cw=clockwise
As loop moves in -y direction, B decreases from I in wire. B from I wire points into the page. To oppose the decrease IN FLUX FROM THE DECREASE IN B, Binduced will point into the page (to try to increase the flux) & current will be clockwise
(a) ccw
(b) cw
(c) no induced current
11/7/2018

15. Demo: E&M Cannon Connect solenoid to a source of alternating voltage.~
side view v B
Connect solenoid to a source of alternating voltage.
The flux through the area ^ to axis of solenoid therefore changes in time
A conducting ring placed on top of the solenoid will have a current induced in it opposing this change.
There will then be a force on the ring since it contains a current which is circulating in the presence of a magnetic field.
Note that it’s the off-axis component of B (the “fringe field”) that flings the ring.
DEMO: 6D-11 Jumping Ring – EM Cannon
11/7/2018

16. Example 4
Suppose two identically shaped rings are used in the demo. Ring 1 is made of copper (resistivity = 1.7X10-8 m); Ring 2 is made of aluminum (resistivity = 2.8X10-8 m). Let F1 be the force on Ring 1; F2 be the force on Ring 2.
Ring 1
Ring 2
(a) F2 < F1
(b) F2 = F1
(c) F2 > F1
11/7/2018

17. Faraday’s Law in terms of E Field
x x x x x x x x x x r E B
11/7/2018

18. Example 5 t B z 1
X X X X
X X X X X X X
X X X X X X X X
X X X X X X X X X x y
The magnetic field in a region of space of radius 2R is aligned with the –z direction and changes in time as shown in the plot.
What is sign [direction] of the induced emf in a ring of radius R at time t=t1?
(a)  < 0
(E cw)
(b)  = 0
(c)  > 0
(E ccw)
11/7/2018

19. Example 6 t B z 1
X X X X
X X X X X X X
X X X X X X X X
X X X X X X X X X x y
The magnetic field in a region of space of radius 2R is aligned with the –z direction (into the page) and changes in time as shown in the plot.
What is the relation between the magnitudes of the induced electric fields ER at radius R and E2R at radius 2R? 2R R
(a) E2R = ER
(b) E2R = 2ER
(c) E2R = 4ER
11/7/2018

20. Faraday’s Law in terms of Electric Fields
Summary
Faraday’s Law (Lenz’ Law)
a changing magnetic flux through a loop induces a current in that loop
negative sign indicates that the induced EMF opposes the change in flux
Faraday’s Law in terms of Electric Fields
11/7/2018

21. I2 is decreasing in magnitude I2 is constant
QUIZ lecture 16
Two current loops are perpendicular to the z axis and are centered on the this axis. Current I2 is clockwise. I1 is the induced current in the upper loop. If I1 is counterclockwise, which statement is true? z
I2 is decreasing in magnitude
I2 is constant
I2 is increasing in magnitude I1 y x I2
11/7/2018

22. I2 is decreasing in magnitude I2 is constant
QUIZ lecture 16
Two current loops are perpendicular to the z axis and are centered on the this axis. Current I2 is counterclockwise. I1 is the induced current in the upper loop. If I1 is counterclockwise, which statement is true? z
I2 is decreasing in magnitude
I2 is constant
I2 is increasing in magnitude I1 y x I2
11/7/2018

23. I1 is decreasing in magnitude I1 is constant
QUIZ lecture 16
Two current loops are perpendicular to the z axis and are centered on the this axis. Current I1 is counterclockwise. I2 is the induced current in the bottom loop. If I2 is clockwise, which statement is true? z
I1 is decreasing in magnitude
I1 is constant
I1 is increasing in magnitude I1 y x I2
11/7/2018

24. I1 is decreasing in magnitude I1 is constant
QUIZ lecture 16
Two current loops are perpendicular to the z axis and are centered on the this axis. Current I1 is clockwise. I2 is the induced current in the bottom loop. If I2 is clockwise, which statement is true?
I1 is decreasing in magnitude
I1 is constant
I1 is increasing in magnitude
11/7/2018

25. QUIZ lecture 16
Suppose a loop is placed in a uniform magnetic field and is spun around a vertical axis as shown. The loop makes one complete revolution every second.
The current induced in the loop
is zero
changes direction once per second
changes direction twice per second
11/7/2018