Videos
SP212
Ch. 30 – Induction
http://digg.com/video/what-happens-when-you-drop-amagnet-inside-a-copper-tube
paper plate speaker http://youtu.be/Awef78YtWmc
Maj Jeremy Best USMC
Physics Department, U.S. Naval Academy
March 22, 2016
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 22, 2016
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Find the Physics
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SP212
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SP212
March 22, 2016
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Michael Faraday
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Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
Induction
Faraday’s Law
In the early 19th Century, Michael Faraday
discovered that a changing magnetic field could
induce an emf (and hence a current) in a loop.
Specifically, he found that a current was generated
when the magnetic flux, ΦB changed. This is now
known as Faraday’s Law:
One easy way to look at this is to imagine the
~ passing through a
number of magnetic field lines B
loop.
Of course, this means we need to Calculate the
magnetic flux.
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 22, 2016
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Magnetic Flux
Faraday’s Law gives us:
Z
ΦB =
~ · dA
~
B
This is very similar to the definition of electric flux:
Z
~
ΦE = E~ · d A
~ is a vector of magnitude dA that is
where d A
perpendicular to (pointing out of) a differential area.
Faraday’s Law is the fundamental operating principle of
transformers, inductors, and many types of electrical
motors, generators and solenoids.
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SP212
March 22, 2016
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Magnetic Flux
Given the definition of a Tesla, that gives us:
For only the special case when B is uniform throughout
the area of the loop, this integral simplifies to:
~ ·A
~ = BA cos θ
ΦB = B
Magnetic flux is measured in a unit called a weber (Wb)
1 Wb = 1 T m2
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 22, 2016
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1 Wb =
Nm
A
which is a force over a distance divided by a current.
The magnitude of the emf E induced in a conducting
loop is equal to the RATE at which the magnetic flux
ΦB through that loop changes with time. Some
authors (and the programmers of the TI calculator),
consider flux more fundamental than magnetic field, and
call the Tesla magnetic flux density, B = Φ/A. This is
not a big deal.
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 22, 2016
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Induction
Induction
If the flux through a loop changes, an emf (and hence a
current) is induced in the loop.
−d(BA cos θ)
−dΦB
=
dt
dt
If there are multiple loops (N of them),
this applies to each loop, and the total is:
dΦB
E = −N
dt
E=
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SP212
March 22, 2016
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Induction
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SP212
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Heinrich Lenz
Because ΦB is a combination of two vectors, there are
three ways that ΦB could change, and hence a voltage
could be induced.
~ could change
The strength of B
~ could change
The size of A
~ and A
~ could change
The angle between B
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SP212
March 22, 2016
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Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
Lenz’s Law
Lenz’s Law
Lenz’s Law is an amazing physics principle.
Mathematically, it is expressed: − . It tells us the
direction of the induced current .
The induced current is in the direction such that the field
produced by the current opposes the change in flux
which produced it.
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 22, 2016
13 / 28
Find the Physics
This can happen in two ways:
1
Current in the loop causes a magnetic field that
opposes the movement of a magnet towards it.
(pole to pole)
2
Current in the loop causes a magnetic field that
opposes the change in flux .
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 22, 2016
14 / 28
The Induced Electric Field
We began this chapter by stating Faraday’s Law:
E=
−dΦB
dt
But recall that potentials are related to electric fields:
I
E = E~ · d~s
I
−dΦB
E~ · d~s =
dt
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SP212
March 22, 2016
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SP212
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Inductors
Inductors
Last chapter, we discussed solenoids, and the fact that
when a current passes through them, they produce a
magnetic field: B = µ0 ni. This magnetic field is
uniform across the cross-sectional area of the solenoid. A
magnetic field and an area are a flux! We define the
inductance of this inductor as
L=
The unit of inductance is the henry (H)
1 H = 1 Tm2 /A
In circuits, you will see inductors symbolized:
NΦB
i
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SP212
March 22, 2016
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Inductors
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SP212
March 22, 2016
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March 22, 2016
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Paper Plate Speaker
So what’s the inductance of a solenoid?
NΦB
i
N = nl
ΦB = BA = µ0 niA
L=
http://youtu.be/Awef78YtWmc
L = µ0 n2 lA
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 22, 2016
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Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
Inductors
Inductors
So what’s the point of an inductor? Well, the definition
of inductance tells us NΦB = Li. And Faraday’s law
connects flux and emf:
This tells us the most important thing about inductors:
An inductor acts so as to resist a change in current.
−d(NΦB )
dt
−d(Li)
E=
dt
di
= −L
dt
E=
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SP212
March 22, 2016
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Inductors in Circuits
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Inductors
KVL gives us the
solution:
R
− iR − L
E
L
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
di
+E =0
dt
Note the similarity to
the capacitor
equations we worked
with earlier. The
results are also similar.
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i = (E/R)(1 − e −t/τ )
τ = L/R
i = (E/R)e −t/τ
An inductor stores energy in its magnetic field:
UB = (1/2)Li 2
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SP212
March 22, 2016
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Energy Storage in Fields
Mutual Inductance: Transformers
With this new statement of energy storage by inductors,
we can deduce the energy density the device stores. Take
the total energy stored in the field and divide by the
volume of the inductor, LA.
1 1 2
B
2 µ0
1
uE = 0 E 2
2
uB =
If the flux through one inductor(coil) is produced by the
field of another inductor, the voltage induced in coil 2 is
related to the current change in coil 1 (and vice versa).
We defined the concept of inductance as L21 i1 = N2 Φ21
. Differentiating with respect to time we find
L21
di1
dΦ21
= N2
= −E2
dt
dt
These ideas will be important when we talk about
electromagnetism detached from circuits.
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SP212
March 22, 2016
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SP212
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Mutual Inductance: Transformers
One can show that the inductance of 2 because of 1 is
the same as the inductance of 1 because of 2. This is
not obvious or easy. We call this single quantity the
mutual inductance, M = L21 = L12 .
Wiley Plus Homework
Chapter 30: Questions 1, 4, 7, 9, 11. Problems: 3, 7, 15,
29, 32, 34, 36, 41, 42, 44, 45, 50, 53, 58, 62, 65, 68, 69,
81, 94. Yeah, its a lot.
di1
dt
di2
E1 = −M
dt
E2 = −M
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SP212
March 22, 2016
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SP212
March 22, 2016
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