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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.
dt
d B ||dt
dN B ||
If coil has N turns
Change in flux may be due to
• Change in magnetic field• Change in the area• Both.
AdBB
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.
dt
d B
Motional EMF
Induced current flows in the loop
External agent pulls the loop with constant speed
BAB
BDxB
dt
d B ||
BDv||
R
BDv
RI ind
||
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
vFP 1DBvIP ind
R
vBDP
222
• 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.
• Find Induced emf = BLv = 0.619 V
• Current in the conducting loop.
• I = /R = 1.49 A
•Assume B = 1.18 T, L = 10.8 cm, v = 4.86 m/s resistance of rod as 415 m
•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.ndt-ed.org/EducationResources/HighSchool/Electricity/eddycurrents.htm
• http://www.ndt-ed.org/TeachingResources/NDT_Tips/LenzLaw.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.
0EDiv
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.
sdE
dt
dsdE B
dt
BdECurl
adBdt
dsdE
In the static case, Faraday’s law reduces to
dt
BdECurl
0ECurl
0 sdE
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)
adBdt
dsdE
r
adBdt
dsdE
2)(2 rtBdt
drE
dt
dBrrE 22
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?
dt
dBasdE 2
ds
B
Torque on the segment ds
RsdE
Rdt
dBa 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.
2/322
20
2 xR
RIB
3
20
2 x
RIB
23
20
2r
x
RIBAB
Compute the emf generated in the smaller
loop
• Direction of current is anticlockwise as seen from above.
23
20
2r
x
RIBAB
vrx
RI
dt
d B 24
20
2
3
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.
2tan2 xA
2tan2 BxBAB
2tan2 2 tBv
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?
s
aa
a
Flux through the loop
s
aa
a
adyy
Ias
s
B
2
0
s
asIaB ln
20
• Direction of induced current is anticlockwise.
s
asIaB ln
20
dt
ds
sdt
ds
as
Ia 11
20
vass
Ia
)(
1
2
20