Download L10_EM_Induction

A changing magnetic field (flux) can create an emf (DV)
 Demo
Recalling magnetic flux
 
 E   E  dA   E cos( )dA
Recall electric flux: General expression
A
Uniform field, flat surface
 
 B   B  dA   B cos( )dA
 Magnetic flux is: General expression
A
Uniform field, flat surface
General
A
 
 E  E  A  EA  E ( A cos  )
A
 
 B  B  A  BA  B( A cos  )
Uniform field, flat surface
Recalling Gauss’s Law for magnetic flux
As we have seen, magnetic forces come from electric charges in motion.
There are no free magnetic charges. Magnetic field lines diverge from N
poles and converge into S poles, but they do not begin or end at either pole.
Then Qmagnetic = 0, so that there cannot be enclosed charge. Gauss’s Law for
magnetism is then:
 
 B   B  dA  0
A
We have encountered magnetic flux before, and this is one of its
important properties. Still, this law does not have a lot of direct
applications, but Faraday’s Law of Induction, introduced in this
section, does!
Also recalling Faraday. (Cage, the Farad, etc.) There’s more!
From Wikipedia:
Michael Faraday, (1791 – 1867) was an English
chemist and physicist who contributed significantly to
the fields of electromagnetism and electrochemistry. He
established that magnetism could affect rays of light
and that there was an underlying relationship between
the two phenomena.
Some historians of science refer to him as the best
experimentalist in the history of science. It was largely
due to his efforts that electricity became viable for use
in technology. The SI unit of capacitance, the farad, is
named after him. Faraday's law of induction states that
a magnetic field changing in time creates a proportional
electromotive force.
Faraday’s Law of Induction
The Law:
If the magnetic flux through a closed loop is changing with
time, an emf is generated around the loop. More precisely:
Voltage drop, DV,
around the loop
d B
emfloop  
dt
Flux passing
through the loop
(We’ll discuss the minus sign a bit later.)
If there are N turns of wire in the loop,
each sees the same changing flux. Then:
A circuit will respond to this emf the same
way it would to a battery (with no internal
resistance). In the loop at right, if the
magnetic flux inside the loop is changing,
then: emfloop = DV = RI. Discuss “where”.
emfloop   N
I
d B
dt
R
Making magnetic flux through a loop change with time
1. Increase or decrease the
strength of the magnetic field
through stationary loop.
2. Move a non-uniform field source
toward or away from the loop.
3. Move the loop in a non-uniform
field.
4. In a uniform field, change the
area of the loop.
5. All of the above, in any
combination.
When motion is involved, the
result is called “motional emf”.
(emo?)
 Demo
Basic alternating current (A.C.) generator
In a basic A.C. generator, a permanent magnet provides a reasonably uniform
magnetic field. As the generator loop turns in this field, the flux through the loop
changes sinusoidally with time. This causes the output voltage (induced emf) to
also change sinusoidally, 90 degrees out of phase with the flux. We can see
why this happens by applying Faraday’s Law of Induction to the flux:
Let:
 B   0 cos(t )
Then:
d
[ 0 cos(t )]
dt
  0 sin( t )
emf  
Motional emf
 Demo
Basic D.C. Generator
Like the D.C. motor, the D.C. generator has a split-ring commutator. This
swaps the direction of the induced emf whenever it would go negative.
Note: Basic D.C. motors and generators have the same construction, and
are interchangeable.
As you can see, the basic generator produces a very “coarse”, wavy, D.C.
output voltage. The output can be smoothed with more loops (eg. 3) and
more splits in the commutor ring (eg. 6). It can smoothed further by the use
of diodes.
More motional emf
More realistic. Use to
discuss energetics.
Lenz’s Law
The direction of any magnetic
induction effect is such as to
oppose the cause of the effect.
This may sound a bit mysterious, but it is actually an easy way to
figure out the direction of induced magnetic fields and induced
currents without having to go through a chain of right-hand rules.
Lenz’s Law: looking at cases
More generally: the sign of the induced emf
A solenoid inside a loop.
If the outer loop is turned into a solenoid, we can
understand the demonstration with nested solenoids.
The Faraday disk dynamo
 Calc
A loop entering, then leaving, a region of
uniform magnetic field
R
Consider (1) the induced emf and current in the loop, (2)
forces on the loop as it travels from -2L to 2L, (3) work
required to maintain constant v, and energy flow.
 Calc
Eddy currents and damping
Use analysis from previous slide to
explain what is happening here.
Metal detection using eddy currents
Induced electric fields: Faraday’s Law in terms of fields
Again, the Law:
emfloop  
d B
dt
Faraday’s Law of Induction, as we’ve used it so far, looks like it’s meant to be
applied to voltage drops only. But let’s analyze the emf around the loop in
terms of electric field. Recall that DV along a path is the line integral of E. We
can make this path into a loop, and rewrite the left-hand side as:
emfloop
 
d B
  E  ds  
dt
I
E
R
For example, in the circuit at right, we see that there
is an electric field circulating the loop.  Not path
independent any more! Try both directions.
Now imagine erasing the circuit and using a “calculational loop”. This equation
also applies to free space, and says that a changing magnetic field can create an
electric field. This is “half” of what’s needed for electromagnetic waves!
Staying on a theoretical roll…
We’ll be getting back to circuits, and a new circuit
element, the “inductor”, in a few slides. But we’re at the
point where, with one modification of Ampere’s Law, we
will have all four of Maxwell’s Equations in hand.
These, together with the Lorenz Force Law, completely
specify electromagnetism. So we’ll do this first.
James Clerk Maxwell, 1831-1879
Scottish mathematician and theoretical physicist.
His most significant achievement was formulating
a set of equations that for the first time expressed
the basic laws of electricity and magnetism in a
unified fashion. He also developed the Maxwell
distribution, a statistical means to describe
aspects of the kinetic theory of gases. These two
discoveries helped usher in the era of modern
physics, laying the foundation for future work in
such fields as special relativity and quantum
mechanics.
“[The work of Maxwell]...[is] the most profound
and the most fruitful that physics has experienced
since the time of Newton.”—Albert Einstein, The
Sunday Post
Maxwell’s addition to Ampere’s Law
Recall Ampere’s Law:
 
 B  ds  0 I encl
As Maxwell did, we consider a capacitor
being charge by a current iC. First,
construct an Amperean loop around the
wire and notice that we can find the B
field around the loop from the current
passing through the plane surface. But,
mathematics allows us to distort the
Amperean surface membrane until it
passes through the gap of the capacitor.
In this region, there is no current flowing,
but instead there is a changing electric
field. Maxwell realized that the second
surface must give the same answer, and
added a term to Ampere’s Law.
Including this new term, we
get the Ampere-Maxwell Law:
The charge on the capacitor is:
0 A
( Ed )   0 AE   0  E
d
“Current” through the capacitor is then:
d E
dQ
iD 
 0
dt
dt
Q  CV 
 
d E 

B

d
s


(
i

i
)


i



0 C
D
0 C
0

dt


Maxwell’s Equations
Gauss’s Law for E
  Qenclosed
 E  dA 
A
Gauss’s Law for B
0
 
 B  dA  0
A
Faraday’s Law of Induction
Ampere-Maxwell Law
 
d B
 E  ds   dt
 
d E 

 B  ds   0  iC   0 dt 
Notice, from the 3rd and 4th equations, that changing magnetic flux can
create electric fields and changing electric flux can create magnetic fields.
This can happen even when there are no charges or currents present, and
leads to the creation of electromagnetic waves  light!
Magnetic field inside a charging capacitor
 Find