Transcript Document

Induction
II
Law of Induction
• The magnitude of the induced emf in a
circuit is equal to the rate at which the
magnetic flux through the circuit is
changing with time.
d B
| | 
dt
dB
| |  N
dt
If coil has N turns
 
 B   B  dA
Change in flux may be due to
• Change in magnetic field
• Change in the area
• Both.
Lenz’s law
• The flux of the magnetic field
due to the induced current
opposes the change in the flux
that causes the induced current.
d B
 
dt
Motional EMF
External
agent
pulls the
loop with
constant
speed
Induced current flows in the loop
B  BA
B  BDx
d B
| | 
dt
|  |  BDv
I ind
|  | BDv


R
R
F1 is the net magnetic force
• If external agent pulls with
constant speed
• Fext = F1 = Iind DB
• Mechanical power
P = F1 v
The power expended by
the external agent
P  F1v
P  I ind DBv
2
2 2
D B v
P
R
• A conducting rod of length L is being pulled
along horizontal, frictionless and conducting
rails. A uniform magnetic field fills the region
in which the rod moves. Assume B = 1.18 T,
L = 10.8 cm, v = 4.86 m/s, resistance of
rod as 415 m.
•Assume B = 1.18 T,
L = 10.8 cm, v = 4.86
m/s resistance of rod
as 415 m
• Find Induced emf
•  = BLv = 0.619 V
• Current in the conducting loop.
• I = /R = 1.49 A
•At what rate does the internal
energy of rod increase?
•P =  Iind = 0.922 W
•Force that must be applied by
external agent to maintain its motion
•F = ILB = 0.190 N
•At what rate does this force do
work on rod?
•P = F v = 0.922 W
Eddy Currents
 An emf and a current are induced
in a circuit by a changing magnetic
flux.
 When the magnetic flux through a
large piece of conductor changes,
induced current appear in the
material in small loops.
 These are called eddy currents as
they induce in little swirls/eddies.
• http://www.ndted.org/EducationResources/HighSchool/Ele
ctricity/eddycurrents.htm
• http://www.ndted.org/TeachingResources/NDT_Tips/Lenz
Law.htm
Eddy currents and energy
loss
• They can increase internal energy and
thus temperature of the material
• Big eddy currents
larger energy
loss
• Materials which are subjected to
magnetic fields are often constructed in
many small layers.
Eddy currents slow down the
motion of the conductor
A cylindrical bar magnet is dropped down
a vertical aluminum pipe of slightly large
diameter . It takes several seconds to
emerge at the bottom, whereas, identical
piece of unmagnetized iron makes the trip
in a fraction of a second. Explain why
magnet falls more slowly??
Ans: delay is due to forces exerted on the
magnet by induced eddy currents in the
pipe.
•Advantage
Heating effect can be used
in induction furnace.
Magnetic field cannot force a
stationary charge to move. Then why
the charges move?
Why there is an induced current?
Induced electric fields
A changing magnetic field
induces an electric field.
•Induced electric field exists,
even when ring is removed.
It is always tangential.

DivE  0
Some facts
• The driving force for induced currents
is induced E-field
• It exists, even when ring is removed.
• It has no radial component.
• As real as that might be setup by a real
stationary charge.
 
   E  ds
 
d B
   E  ds  
dt
 
d  
   E  ds    B  da
dt


dB
Curl E  
dt


dB
Curl E  
dt
In the static case, Faraday’s law
reduces to

Curl E  0
 
E

d
s

0

You can not define a potential for an induced
electric field.
A uniform magnetic field B(t) pointing
straight up fills the shaded circular
region. If B is changing with time what
is the induced electric field ?
B(t)
 
d  
   E  ds    B  da
dt
r
 
d  
 E  ds   dt  B  da

d
2
E 2r  
B(t )r
dt
dB
E 2r  r
dt
2

If B is increasing
with time,
induced current
will run
clockwise as look
from above.
A line charge  is glued onto the rim of a
wheel of radius R, which is then suspended
horizontally . It is free to rotate. The spokes
are made of wood. In the central region out to
radius a there is a uniform magnetic field
pointing up. Now someone turns the field off.
What happens?
B
ds
 
2 dB
 E  ds  a dt
Torque on the segment ds
 
    E  ds R
dB
  a
R
dt
2
Two parallel loops of wire are
shown with common axis.
Smaller loop is above the
larger loop by a distance
x>>R. Magnetic field due to
current i in the larger loop is
constant through the smaller
loop and equal to the value on
the axis. Suppose x is
increasing with constant rate.
(a) Determine the flux across the
area bounded by smaller loop as
a function of x.
B
0 I
2
B
R
R
2
2

2 3/ 2
x
0 I R
B  BA 
2 x
2
3
0 I R
2 x
2
3
r
2
Compute the emf generated in
the smaller loop
B  BA 
0 I R
2 x
2
3
r
2
d B 3 0 I R
2
 

r v
4
dt
2 x
• Direction of current is anticlockwise as
seen from above.
2
Two straight conducting
rails form an angle 
where their ends are
joined. A conducting bar
in contact with the rails
and forming an isoscale
triangle with them, starts
at the vertex at time t = 0
and moves with constant
velocity v to the right. A
magnetic field points out
of the page.
Find emf
induced as a
function of time.
A  x tan
2
2
B  BA  Bx tan 2
2
  2 Bv t tan
2
2
A square loop of wire lies on a table, a
distance s from a very long straight wire,
which carries a current I. If someone
pulls the loop away from the wire at
speed v, what emf is generated?
a
a
s
a
Flux through the loop
a
a
s
a
B 
sa

s
0 I
ady
2y
 0 Ia  s  a 
B 
ln

2  s 
 0 Ia  s  a 
B 
ln

2  s 
 0 Ia  1 ds 1 ds 
 


2  s  a dt s dt 
0 Ia
1

v
2 s(s  a)
2
• Direction
of
anticlockwise.
induced
current
is