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Two questions:
(1) How to find the force, F on the electric charge, Q excreted by the


field E and/or B?
 
F  qE  qv  B
(2) How fields E and/or B can be created?
Maxwell’s equations
Gauss’s law for electric field
Electric charges create electric field:
 E   EA cos  Q /  0
For one not moving
(v<<c) charge:
Ek
Gauss’s law for magnetic field
Magnetic charges do not exist:
 B   BA cos  0
Q
r2
Amperes law
Faraday’s law
Electric current creates magnetic field: A changing magnetic field induces an EMF
 Bl cos  0 I
(As we will see later,
this law should be extended)
 B
 
t
A changing magnetic flux induces an
electric field !
8. Electromagnetic induction
(Faraday’s law)
2) EMF (review)
1) Flux (review)

 B  BA cos 
 B   BA cos 
Units: (weber)
 B
 
t
 
  BA cos  
t
A changing magnetic flux
induces an EMF
 B - flux through a closed loop

   El cos 
[ΦB] = 1 Wb = 1 T·m2
3) Faraday’s law
- EMF in the closed loop
For N loops :  N  N 1
W
Fl cos 


q
q
 El cos   
 BA cos  
t
A changing magnetic flux induces an
electric field !
This electric field is not a potential field.
The field lines form a closed loops.
4) Lenz’s law
The direction of any magnetic induction effect is such as to oppose the cause
of the effect
For instance: a current produced by an induced emf moves in a direction so
that its magnetic field opposes the original change in flux

v
S
N
I

B

v
S
N
I

B

v
N
S
I

B

v
N
S

B
I
Example: If a North pole moves toward the loop in the plane of the page,
in what direction is the induced current?
Since the magnet is moving parallel to the loop,
there is no magnetic flux through the loop.
Thus the induced current is zero.
Example: In order to change the magnetic flux through the loop,
what would you have to do?
1)
2)
3)
4)
5)
drop the magnet
move the magnet upwards
move the magnet sideways
all of the above
only (1) and (2)
Moving the magnet in any direction would change the
magnetic field through the loop and thus the magnetic flux.
1)
2)
3)
4)
5)
tilt the loop
change the loop area
use thicker wires
all of the above
only (1) and (2)
Since,  B  BA cos
changing the area or tilting the loop (which varies the
projected area) would change the magnetic flux through the loop.
Example: Wire #1 (length L) forms a one-turn loop, and a bar magnet is dropped
through. Wire #2 (length 2L) forms a two-turn loop, and the same magnet is
dropped through. Compare the magnitude of the induced currents in these two
cases.
1)
2)
3)
4)
I1
I1
I1
I1
>
<
=
=
I2
I2
I2  0
I2 = 0
 B
 B1
 
 N
t
t
S
N
S
N
Induced emf is twice as large in the wire with 2 loops. The current is given by
Ohm’s law: I = V/R. Since wire #2 is twice as long as wire #1, it has twice the
resistance, so the current in both wires is the same.
Example: A bar magnet is held above the floor and dropped. In 1, there is
nothing between the magnet and the floor. In 2, the magnet falls through a
copper loop. How will the magnet in case 2 fall in comparison to case 1?
1) it will fall slower;
2) it will fall faster;
3) it will fall the same
When the magnet is falling from above the loop in 2, the
induced current will produce a North pole on top of the
loop, which repels the magnet.
When the magnet is below the loop, the induced current
will produce a North pole on the bottom of the loop,
which attracts the South pole of the magnet.
S
N
S
N
Example 1: A coil of 600 turns with area 100 cm2 is pleased in a uniform
magnetic field. The angle between the direction of the field and perpendicular
to the loop is 60°. The field changes at a rate of 0.010 T/s. What is the
magnitude of the induced emf in the coil?
N  600
 B   BA cos   NBA cos 
  60
 B
B
N
A cos 
t
t
A  100cm 2
B
 0.010T / s
t
 ?
 B
B
 
N
A cos 
t
t
  600  0.010T / s   100 104 m 2 cos 60  0.03V
 B   BA cos   NBA cos 
Example 2:
 B
A
 NB
cos 
t
t
N  600
  60
A
 100cm 2 / s
t
B  0.010T
 ?
 
 B
A
 NB
cos 
t
t
  600  0.010T   100 104 m 2 / s cos 60  0.03V
Example 3: A 12.0-cm-diameter wire coil is initially oriented perpendicular
to a 1.5 T magnetic field. The loop is rotated so that its plane is parallel to
the field direction in 0.20 s. What is the average induced emf in the loop?
N 1
f   0
 in   90
2r  12.0cm
 B  BA cos 


 B   Bf    Bin   BA cos 0  cos 90  BA
 
 B BA

t
t
A  r 2
B  1.5T
t  0.20 s
 ?
2

1.5T     0.12m / 2

0.20s
 8.5 10  2 V
1) Rotating loop: 
8a. Applications of Faraday’s law
 t
  2f  2 T
I
 B  BA cos  BA cos t
B
 B
BA cos t 
 

t
t
  BA sin t
Example:
N  200
B  0.03T
I
 max
 max  NBA
A  100cm 2
  100 s 1
 max  ?
 max  200  0.03T   100  10 4 m 2  100s 1   6V
Example: A generator rotates at 60 Hz in a magnetic field of 0.03 T.
It has 1000 turns and produces voltage that is 120 V at a pick.
What is the area of each turn of the coil?
f  60 Hz
B  0.03T
N  1000
 max  120V
A?
 max  NBA
  2f
 max
A
2fNB
120V
A
 10 2 m
2  60 Hz   1000  0.03T 