Magnetic flux and Faraday’s Law
Other way to show:
So far:
ÖMagnetic fields arises by moving charges
A coil experiences a current when magnetic passing through it
varies
Now: Changing the magnetic field arise electric field
(Michael Faraday conducted an experiment to prove this)
At the moment one closes the
switch ammeter deflect and
returns immediately to zero.
Again deflection at the moment
opening the switch
The change in the magnetic field is described with the change in
magnetic flux
There is no contact between the
coils. We called this
phenomenon a induce current.
If we induce a current we have to have a potential difference:
induced emf (electromotive force)
Electric Flux
)el = E A cosT
Magnetic Flux
)mag = B A cosT
[Tm2]=[Wb, Weber]
In detail:
The magnitude of the induced current/ induced emf is proportional
to the change of the magnetic field.
ÖThe more rapidly the change in the magnetic field the more
the higher the induced current/emf
Faraday’s Law of Induction:
Induced emf is proportional to the rate of flux change
emf
N
')
't
Magnitude: emf
) final ) initial
I final I initial
')
N
't
N
Example:
Lenz Law (expresses the meaning of the minus)
An induced current always flow in a direction that opposes the
change that caused it.
Motional emf: Qualitative
Field is constant but the magnetic flux changes because rod
accelerates due to gravity.
Öinduce emf because of motion
Lenz’s Law: A decrease in the surface causes a decrease in
magnetic flux.
Therefore the magnetic field from the induced current has to
strengthen the field. The current flows counterclockwise.
ÖCurrent through rod causes a force against gravity. If Fmag =
Fg then rod moves with a constant velocity
Motional emf: Quantitative
Power
F = I l B sinT = Bvl/R lB = B2 l2 v/R
Pmech= F v = (B2 l2 v/R) v = B2 l2 v2/R
Pelectric= I2 R = (Bvl/R)2 R = B2 l2 v2 /R
Application: Generator
No Fg in this case. We move the rod to the right side with
a constant external force Fext.
ǻ)mag = B ǻA
with
A = v ǻt l
ǻ)mag= B v ǻt l
Faradays Law to calculate emf
emf
N
')
't
N
Bvl't
't
N=1 (1 loop)
|emf| = Bvl
Current through light bulb
I = |emf|/R = Bvl/R
Relationship between E-field and B-field
V= El emf = Bvl
El = Bvl
or E = B v
Definition of inductance L
')
'I
')
emf N
L
and L= N
[H, Henry = Vs/A]
't
't
'I
Inductance
Coil – coil induction is referred to as mutual induction.
Example: Inductance of a solenoid
A coil with a changing current/magnetic field can induce a
current in itself: self inductance
Example:
Coil in electric circuit
(or inductor in short resist
changes in its electric current)
By closing the switch it is not
possible for current to rise
immediately to final value.
The inductor slows the
process.
(Remember capacitor)
Capacitor IJ = RC
Inductor
IJ = L/R
I
Decreasing current: Inductor try to keep current constant.
ĭ ~ I it must be that
N
RL Circuit
Switch open:
Break in the circuit.
Switch closed:
Initially current is zero.
Current steadily increase with
time.
ÖMagnetic flux increase with
time
ÖInduce an emf (Faraday) in
the opposite direction to the
current (Lenz)
ÖResistance against I
ÖWork is done!!
Self-induced emf
')
because
emf N
't
')
BA 0
(P ( N / l ) I ) A
N
N 0
'I
I 0
I
or N / l n L P 0 A n 2l
L
') 'I
~
't 't
emf
(1 e t / W )
R
emf
(1 e tR / L )
R
P0 A N 2 / l
Work is required to
establish a current through
a inductor. This is a
conservative process and
the work/energy is stored
in the magnetic field,
similar to capacitor.
An application: Transformers
Changes voltage from one value to the other.
In principle:
Magnetic flux in a primary coil induces a voltage in a secondary
coil.
Pav= ½ I emf = ½ I (LI/t)
t: time of increasing
current
Pav= ½ L I2 /t
Energy stored: U = Pavt
UB = ½ L I2
Energy density uB = magnetic energy / volume
Example: Solenoid
L
P 0 A n 2l
B
P 0 nI
U
1
2P0
U
1
P 0 A n 2l I 2
2
B 2 Al
energy density u B
1 2
B
2P0
')
't
')
emf s N s
't
if the number of coils N is different, voltage is different.
Rearrange for ĭ give
emf p
N p
Transformer equation
emf p 't
Np
emf p
Np
emf s
Ns
emf s 't
Ns
or
emf p
emf s
Np
Ns
P= V I = emf I
Energy conservation:
The higher the voltage the lower the current
emf p
emf s
Np
Ns
Is
Ip