known that magnetic field was produced by electric field. power note 9: faraday’s...
TRANSCRIPT
Note 9: Faraday’s Law
In 1831, Faraday discovered that EMF (electromotive force, i.e., voltage)
was induced by time varying magnetic flux. This was a monumental
discovery in the physics history. Before Faraday, various attempts had
been made to produce electric field from magnetic filed since it was
known that magnetic field was produced by electric field. Power
generation at large power plants and wind power generation all
depends on the Faraday's law.
Photocell produces electricity directly from solar radiation. This is
quantum mechanical effect.
(a) North pole of bar magnet is approaching a coil. The magnetic
flux in the coil is increasing with time. The current driven by the
induced EMF tends to oppose the increasing ma .
(
gne
b)
tic fl
No
ux
motion no EMF
(c) When the north pole is moving away from the loop,
EMF opposite to the case (a) is induced. The direction of the current
is reversed.
EMF induced by Time Varying Magnetic Flux
Faraday's law can be expressed as
Electromotive force =
where is the magentic flux (Wb).
Change in the magnetic flux is due to either timevarying
magnetic field , change in th
A
dEMF
dt
d
t
B A
Be area , or change in
the angle between and .
The magnetic flux through an area is
= cos (Wb)
where is the angle between the B field
and the vector normal to . Change in the
flux can
dA
dt
A
AB
A
B A
be due to that in , , or .B A
2 . When a magnetic field normal to a square coil of 18 cm
having 200 winding turns increases from 0 to 0.5 T in 0.8 sec,
what is the induced EMF?
Sol. The magnetic flux linked to the coil is
B
Example 1
2 4 2
The mi
0.5 T cos 200 18 10 (m ) 1 4.0 (Wb)
0.8 sec
4.0 V nus sig. Note cos 0 1.
A r
n is w.r.t. the di
ectangula
rection of
the current to produce the B field.
NAB t t
dEMF
dt
Example 2.
r coil of area is rotating in a magnetic field
at angular frequency rad/sec. Find the EMF to be induced.
Sol. The magnetic flux through the coil is changing with time as
= cos (Wb)
wh
A ab
B
abB t
ere 0 is chosen when the coild plane is normal to B (maximum flux).
Induced EMF is ( cos ) sin
This is the principle of ac generator.
t
dEMF abB t abB t
dt
A conductor loop of radius ia approaching a current carrying
wire at a velocity . Determine the direction of current induced in the loop.
Sol. The current is the loop will flow so as to o
a
v
Example 3.
ppose the change in the
magnetic flux enclosd by the loop. The magnetic flux out of the page increases
as the loop approaches the wire. Therefore, the current will flow clock wise.
If the coil is moving away from the current, the EMF is reversed.
v
I'
I
0
A rectangular loop of area is moving away from a
current carrying wire at velocity . Find the EMF to be indued.
Sol. The magnetic field due to the current carrying wire is
2
Th
ab
v
IB
r
Example 4.
0 0
0
e magntic flux through the loop is
1 = ln
2 2
Then the induced EMF is
1
2
The secondary current ' in the loop would
flow clockwise. (Its magni
vt a
vt
Ib Ib vt adr
r vt
Ibd aEMF
dt a vt t
I
tude depends on the resistance
in the loop.)
b
a
v
vt
a + vt
I
I '
In the figure, a conducting bar is
moving across the magentic field
at a vlocity . The magnetic flux
is increasing and
the induced EMF is
(V)
The minus sign indicates that the
curre
v
lxB lvtB
EMF lvB
nt fill flow so as to oppose
the change in the magnetic flux,
i.e., counterclockwise in this
case. This is known as Lenz's law.
Motional EMF
5. Electromagnetic Braking. A conducting bar of mass moves across
a magnetic field at an initial velocity on frictionless rails. The total resistance of
the circuit is . When the extern
i
m
B v
R
Example
al force is removed at 0, show that the
velocity decreases expoenentially with time.
Sol. The EMF is ( ) and the current can be found from
0
The equation of motion for the bar gives
t
Blv t I
Blv t RI
2
2
Solution is exp , (sec)i
lBdvm IlB v
dt R
t mRv t v
lB
.
A conductor rod of length is
rotating in a magnetic field at an agular
velocity (rad/sec). Calculate the EMF
induced.
Sol. At radius , the electric field is
(V/m)
Then, the
r
l
r
E vB rB
Example 6
2
0
2
EMF is
1 (V)
2
If a conducting disk of radius is rotating
in a magnetic field, an EMF is induced
between the center and the edge,
1 (V)
2
This is the principle of homopol
l
rEMF E dr Bl
a
EMF Ba
ar generator.
. Problem 25. 0.5T, 1.2 m, 1 , 25 , 0.25kg.
Find the terminal velocity .
cosSol. EMF = cos
The source of motion is the gravitational potential
energy which is converted
B l R m
v
lBvlBv RI I
R
Example 7
2
2
2 2
to heat in the resistor .
sin
cossin
sin 1 0.2 9.8 sin 252.8 m/s
cos 1.2 0.5 cos 25
R
dRI mgh mgv
dt
lBvR mgv
R
Rmgv
lB
. Faraday tried to measure motional emf across the River Thames.
The sensitivity of instruments was not high enough and he could not confirm.
If the earth magentic field (nearly downward) is
Example 7
4
4
5 10 T and water flow velocity
is 3 m/sec, what emf is expected across the river width of 150 m?
Ans. emf = 3 5 10 150 0.225 V
. Having learned that an object moving across a magnetic fie
vBl
Example ld should induce
an emf, a student tried to measure it in a car which goes practically normal to the earth
magnetic field. He used a volt meter to measure voltage across the width of the car.
Explain why he failed.
Ans. The magnetic flux enclosed by the car is constant and there should be no emf.
0
9. The current of a long solenoid having winding
density (turns/m) is given by cos . Find the electric field
at distance from the axis of the solenoid.
Sol. The magnetic field in hte
n I t
r
Example
0 0
2 2
0 0 0 0
2
0 0
2 2
0 0 0 0
0 0
solenoid is cos .
Outside the sollenoid
2 cos sin
1sin
2
Inside
2 cos sin
1sin
2
B nI t
r R
drE EMF R nI t R I t
dt
RE I t
r
r R
drE EMF R nI t R I t
dt
E r I t
Generators and Motors
When a coudctor loop rotates in a magetic field, a sinusoidal
voltage is induced. The magnetic flux is = cos .
Induced EMF is
sin
AB t
dEMF AB t
dt
AC Generator
If a commutator is installed, the voltage becomes unipolar.
DC Generator
Practical dc generatorfor smooth dc EMF
Eddy Current
2
In a conductor, electric field drives current through the Ohm's law
(A/m )
where (S/m) is the conductivity. When a conductor plate is placed
near a coil carrying time varying current, electri
J E
c field is induced
and electric current is driven. The current heats the conductor
(Joule heating). Induction kitchen stove uses this principle.
In the pendulum shown, the one without
slits damps faster because eddy current
can flow without being impeded. The slits
prevents eddy current and oscillation
lasts longer.