Download m 0 N 2 A / l

Walker, Chapter 23
Magnetic Flux and Faraday’s
Law of Induction
Michael Faraday, portrait by Thomas Phillips
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.
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Magnetic Flux
For a “loop” of wire (not necessarily circular)
with area A, in an external magnetic field B,
the magnetic flux is:
  B A  BA cos

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Walker problem # 2
A uniform magnetic field of 0.0250 T points vertically upward. Find
the magnitude of the magnetic flux through each of the five sides of
the open-topped rectangular box shown in the figure, given that the
dimensions of the box are L = 31.5 cm, W = 12.0 cm, and H = 10.0
cm.
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Question: Magnetic Field from
loop
•
Current I flows around the loop as
shown, driven by the battery. Inside
the loop the magnetic field generated
by current I is:
1.
2.
Out of screen
Into screen
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Magnetic flux
generated by
current in loop.
• In this case, we choose
clockwise to be positive
current.
• In the plane of this picture,
the magnetic field generated
by this current is into the
picture inside the loop, and
out of the picture outside of
the loop.

















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Examples of Induced Current
Any change of current in primary induces a current in secondary.
The secondary shows a deflection only during the instant just after
the switch is closed or opened.
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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 flux is increasing
• In c) the flux is decreasing
in magnitude.
• In a) and c) the induced
current has opposite sign.
v
v
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Faraday’s Law of Induction
Lenz’s Law
Faraday’s Law: The instantaneous EMF in a circuit
(w/ N loops) equals the rate of change of magnetic
flux through the circuit:

eN
t
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Question: Changing Magnetic Flux
• This is a plot of magnetic flux (Wb=T·m2) through a coil.
• For which interval (a, b, or c) is the change in flux 0?
a
c
b
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Walker, Problem 23-9
• 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? (answer 7.35 V)
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Inductance
• Consider a solenoid of length l with N windings and radius
r (Area A=p r2).
• A current I produces a magnetic field in the solenoid of
B = m0 N I / l
• This produces a total flux through each winding of
 = A B = (m0 N A / l ) I
– Define the Inductance L = (m0 N2 A / l ) (not a length!!)
• If the current in the solenoid changes, there will be an
induced EMF
• e =  N /t =  (m0 N2 A / l ) (I/t)
• e =  L (I/t)
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Inductance
Walker Problem #42
Determine the inductance of a solenoid with 650 turns in a length
of 25 cm. The circular cross section of the solenoid has a
radius of 4.0 cm. [answer: 10.7 mH]
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Example of Inductance
• Consider Solenoid from previous example,
– Radius r = 5cm, length l = 10 cm, 3500 windings.
• L = (m0 N2 Area / l ) = (m0 N2 p r2 / l )
•
2
7 T  m
2 p 0.05m 
L  4p  10
 3500 
A
0.10m
T  m2
 1.21
 1.21H
A
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Transformers
• Microscopic currents in soft iron yoke amplify magnetic
field produced by coils, and contain all magnetic field lines
to “magnetic circuit” formed by iron.
• Both primary and secondary circuits experience the same
magnetic flux for each winding, but have different number
of windings.
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Flux in Transformer
 = flux through a single winding or either coil.
(produced by currents in either or both circuits)
EMF(primary) =  NP /t
EMF(secondary) =  NS /t
eP / eS = NP / NS
Energy conservation requires
eP I P = eS I S
IP / IS = NS / NP
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Walker problem #61
A step-up transformer has 20 turns on the primary coil and 500 turns
on the secondary coil. If this transformer is to produce an output
of 4600 V with a 12 mA current, what input current and voltage
are needed? [answer: 0.3 A, 184 V]
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