Transcript Phy 211: General Physics I

Phy 213: General Physics III
Chapter 30: Induction & Inductance
Lecture Notes
Electromagnetic Induction
•
We have observed that force is exerted on a charge by
either and E field or a B field (when charge is moving):
Fon a charge = qE + qv  B {together this is the Lorentz Force}
•
Consequences of the Lorentz Force:
– A B field can exert a force on an electric current (moving charge)
– A changing B-field (such as a moving magnet) will exert a magnetic
force on a static charge, producing an electric current → this is called
electromagnetic induction
•
Faraday’s contribution to this observation:
–
1.
2.
3.
•
•
For a closed loop, a current is induced when:
The B-field through the loop changes
The area (A) of the loop changes
The orientation of B and A changes
A current is induced ONLY when any or all
of the above are changing
The magnitude of the induced current
depends on the rate of change of 1-3
q
B
v
F
q
F
N
N
S
Moving
charge
B
v
S
Moving
magnet
Magnetic Flux
• Faraday referred to changes in B field, area and orientation
as changes in magnetic flux inside the closed loop
• The formal definition of magnetic flux (FB) (analogous to
electric flux):
f
FB =  B  dA
B
A
When B is uniform over A, this becomes:
FB = BA  cosf
• Magnetic flux is a measure of the # of B field lines within a
closed area (or in this case a loop or coil of wire)
• Changes in B, A and/or f change the magnetic flux
Faraday’s Law: changing magnetic flux induces
electromotive force (& thus current) in a closed wire loop
Faraday’s Law
• When no voltage source is present, current will flow around a
closed loop or coil when an electric field is present parallel to
the current flow.
• Charge flows due to the presence of electromotive force, or
emf (e) on charge carriers in the coil. The emf is given by:
E
ds
i
e =
 Ed
= iR coil
• An E-field is induced along a coil when the magnetic flux
changes, producing an emf (e). The induced emf is related to:
– The number of loops (N) in the coil
– The rate at which the magnetic flux is changing inside the loop(s), or
dF B
d
e =  E  d = -N
= -N BA  cosf )
dt
dt
Note: magnetic flux changes when either the magnetic field (B),
the area (A) or the orientation (cos f) of the loop changes:
dF B
dB
d  cosf )
dF B
dA
dF B
dt
=A  cosf
dt
dt
=B  cosf
dt
dt
=BA 
dt
Changing Magnetic Field
dB
e  -NA  cosf 
dt
A magnet moves toward a loop
of wire (N=10 & A is 0.02 m2).
During the movement, B
changes from is 0.0 T to 1.5 T in
3 s (Rloop is 2 W).
1) What is the induced e in the loop?
2) What is the induced current in the loop?
Changing Area
A loop of wire (N=10) contracts
from 0.03 m2 to 0.01 m2 in 0.5 s,
where B is 0.5 T and f is 0o
(Rloop is 1 W).
dA
e  -NB  cosf 
dt
1) What is the induced e in the loop?
2) What is the induced current in the loop?
Changing Orientation
d(cosf )
e  -NAB 
dt
or
d(cos ωt )
e  -NAB 
dt
A loop of wire (N=10) rotates from 0o to 90o in 1.5 s, B is
0.5 T and A is 0.02 m2 (Rloop is 2 W).
1) What is the average angular frequency, w?
2) What is the induced e in the loop?
3) What is the induced current in the loop?
Lenz’s Law
• When the magnetic flux changes within a loop of wire, the
induced current resists the changing flux
• The direction of the induced current always produces a
magnetic field that resists the change in magnetic flux (blue
arrows)
B
Magnetic flux, FB
i
Increasing FB
B
i
B
Increasing FB
• Review the previous examples and determine the direction of
the current
Operating a light bulb with motional EMF
Consider a rectangular loop placed
within a magnetic field, with a
moveable rail (Rloop= 2 W).
B = 0.5 T
v = 10 m/s
L = 1.0 m
Questions:
1) What is the area of the loop?
2) How does the area vary with v?
3) What is the induced e in the loop?
4) What is the induced current in the loop?
5) What is the direction of the current?
Force & Magnetic Induction
What about the force applied by the hand to keep the rail moving?
• The moving rail induces an electric current and also produces
power to drive the current:
P = e.i = (5 V)(2.5 A) = 12.5 W
• The power (rate of work performed) comes from the effort of the
hand to push the rail
– Since v is constant, the magnetic field exerts a resistive force on the rail:
FNet = Fhand + FB = 0 or Fhand = FB
The force of the hand can be determined from
P
the power: P = F
v  F
=
hand
Fhand
FB
hand
v
12.5 W
Fhand =
= 12.5 N = FB
m
10 s
Generators & Alternating Current
• Generators are devices that utilize electromagnetic
induction to produce electricity
• Generators convert mechanical energy into
electrical energy
– Mechanical energy is utilized to either:
• Rotate a magnet inside a wire coil
• Rotate a wire coil inside a magnetic field
– In both cases, the magnetic flux inside the coil changes
producing an induced voltage
– As the magnet or coil rotates, it produces an alternating
current (AC) {due to the changing orientation of the coil
and the magnetic field}
• Motors and Generators are equivalent devices
– A generator is a motor running in reverse:
Maxwell’s Equations
Taken in combination, the electromagnetic equations are
referred to as Maxwell’s Equations:
ρdV
qenc

1. Gauss’ Law (E)  E  dA =
=
eo
2. Gauss’ Law (B)
eo
 B  dA = 0
dq
d
=  oe o
E  dA
3. Ampere’s Law  B  d = oienc = o

dt
dt
E
  B  d =  oe o 
 dA
t
dFB
d
=-  B  dA
4. Faraday’s Law  E  d = dt
dt
B
  E  d = -
 dA
t
 oe o
2E = oe o
2E
t2
Significance of Maxwell’s Equations
1. A time changing E field induces a B field.
2. A time changing B field induces an E field.
3. Together, 1 & 2 explain all electromagnetic behavior (in a
classical sense) AND suggest that both E & B propagate as
traveling waves, directed perpendicular to each other AND
the propagation of the waves, where:
2

E
2
 E =  oe o 2
t
and
2

B
2
 B =  oe o 2
t
The product, oeo, has special significance: oe o =
or
vwave =
1
o e o
= 2.99x108 ms = c
1
v2wave