Electromagnetic

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Electromagnetic Induction (plus
inductance and magnetic energy)
Lectures 22, 23, 24
UCSD Ph 2B lectures by George M. Fuller
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George M. Fuller 427 SERF gfuller_at_ucsd.edu 822-1
214
Office Hours 200 PM - 330 PM Tuesday 329 SERF
PM
TH 700 PM - 800 PM WLH 2204
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no quiz
Ch 29,30,31
Ch 31,32
Ch 33
problem assignment TBA
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Timeout! . . . We now have all these laws, some
of which are really fundamental and some of which
are not, i.e., they can be derived from
other laws or they are a consequence of them. For
example, Coulombs Law for the electric field
from a point charge (or a spherical charge
distribution) is really simply a consequence of
Gausss Law for electric charge.
So which are the really basic relations which
govern electricity and magnetism?
We will answer this question, but before we do we
have to clear up some misconceptions you may have
about what it is we have done so far. These
may stem from the quasi-historical approach to
the development of the subject presented in our
book. In particular, the book may have led some
of you to think that we have derived the
behavior of electromagnetism. We have not.
Instead, what we have done is present a few
mathematical relations derived from experiment
(summary of experiments) and we have explored the
consequences of these relations in a number of
settings and for various problems.
So now we will concisely summarize ALL of
electricity and magnetism and tell you how you
should think about this subject.
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How to think about electricity and magnetism . . .
The physical things that are measured in a lab
have to do with the forces on (and consequent
accelerations of) charges, i.e., the flow of
momentum. The Lorentz Force Law gives these and,
in so doing, serves to define the electric and
magnetic fields.
Having so defined the electric and magnetic
fields, there are four relations which relate
these to charges and currents. These are
Maxwells Equations
TRUE ALWAYS
The electric flux through a closed surface is
equal to the charge enclosed (divided by a
constant).
The magnetic flux through a closed surface is
zero.
The line integral of the electric field around a
closed path is equal to minus the rate of change
of magnetic flux through any surface which has
the path as its boundary.
The line integral of the magnetic field around a
closed path is equal to the rate of change
of electric flux plus the net current flowing
through any surface which has the path as its
boundary.
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For the limit where all time derivatives
vanish, and currents are steady and unchanging
with time (the electrostatic and magnetostatic
limits), we have
The electric flux through a closed surface is
equal to the charge enclosed (divided by a
constant).
Coulombs Law
The magnetic flux through a closed surface is
zero.
The line integral of the electric field around a
closed path is zero. (electric force is
conservative)
Amperes Law The line integral of the magnetic
field around a closed path is equal to the net
current flowing through any surface which has the
path as its boundary.
Biot-Savart Law
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Faradays law (the third of Maxwells
equations) tells us that a changing magnetic
field creates an electric field
UCSD Ph 2B lectures by George M. Fuller
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Electromagnetic Induction Faradays Law
Now we let the electric and magnetic fields
change with time. The first phenomenon that we
shall explore is how a changing magnetic flux
through a loop causes an nonzero line integral of
electric field around that loop. If we replace
our arbitrary loop with a real wire or circuit,
then this nonzero line integral of electric
field corresponds to an EMF. This phenomenon is
summarized in Faradays Law.
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EMF around a fixed circuit is the component of
force per unit charge that lies along the wire,
integrated around the entire circuit.
This is the total work done on a unit charge that
goes around the circuit.
Clearly, the electric field induced in this way
and the resulting electric force is NOT
conservative that is, the work done by this
field in the motion of an electric charge between
two points in space can depend on the path taken
between those points.
If there is a real conductor (a wire or whatever)
that is in the shape of the circuit (or loop)
then the induced EMF results in a current that is
related to the total equivalent resistance in
the circuit by Ohms law
If there is no conducting body coincident with
our circuit loop, then there will be no current,
but there will nonetheless be a nonzero line
integral of electric field around the circuit
path (loop).
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Faradays Law (the third of
Maxwells equations)
This minus sign constitutes Lenzs Law.
The physical meaning of this minus sign (Lenzs
Law) direction of induced EMF

and associated current

(if
any). A current induced by a changing magnetic
flux is in the direction which produces a
magnetic field which opposes the sense of change
of the magnetic flux. Energy conservation
demands this minus sign power dissipated by a
current from an induced EMF must be compensated
by a positive rate of work done in changing the
magnetic flux.
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Example Problem 27, Chapter 31 Consider a pair
of conducting rails a distance l apart in a
uniform magnetic field B, which points into the
screen. A resistance R is connected across the
rails, and a conducting bar of negligible
resistance is being pulled along the rails with
velocity v to the right. (a) What is the
direction of the current through the resistor?
(b) At what rate must work be done by the agent
pulling the bar?

• There are two ways to approach this problem
• use Faradays (and Lenzs) law
• use the Lorentz force law

uniform magnetic field into screen
First let us solve it with Faradays law take as
our loop the circuit made by the resistor, the
segments of the rails to the left of the
crossbar, and the crossbar. Lay down a coordinate
system as shown.
v
R
y - axis
rate of change of magnetic flux through our loop
x - axis
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The B-field encircles a long straight wire with a
direction given by the right hand rule and a
magnitude that drops inversely like the distance
from the wire
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Now let us analyze this same problem using just
the Lorentz force law
The charge carriers in the crossbar will feel a
force directed upward, in the positive y-direction
The EMF in the circuit will be the integral of
this force, per unit charge, along the crossbar
length l
This agrees with what we got from Faradays law
a counter clockwise directed current and a
power dissipated in the resistor
By energy conservation this has got to be the
rate of work done by an external agent in pulling
the crossbar to the right.
The crossbar is a current-carrying straight
segment in a magnetic field so it is subject to a
force
This must be countered by a force in the
positive x direction
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Example alternating current generator (AC
generator)
Rotate loop of wire at angular frequency w in
a fixed, uniform magnetic field B.
uniform B-field (into screen)
R
Therefore, at an instant in time t the magnetic
flux through the loop is
and the rate of change of magnetic flux through
the loop is
By Faradays law the induced EMF is
and the alternating current is
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Example AC generator continued . . .
Rate of mechanical work required to turn the
generator is, of course, equal to this. Recognize
that the loop of current-carrying wire
constitutes a magnetic dipole, with a dipole
moment equal in magnitude to the product of the
area of the loop and the current.
There is a torque on this magnetic dipole in the
magnetic field
Rate at which mechanical work must be done to
keep the loop rotating
This is the same power we got above. Remember
that the current is
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Example Problem 26, Chapter 31 A generator
consists of a rectangular coil 75 cm by 1.3 m,
spinning in a 0.14 T magnetic field. If it is to
produce a 60 Hz alternating EMF with peak value
6.7 kV, how many turns must it have?
We will borrow the previous result, but note that
here the EMF will be N times (number of turns)
larger.
solve for number of turns
Now note that the relationship between frequency
and angular frequency is
The number of turns is
UCSD Ph 2B lectures by George M. Fuller
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Example The Solenoid. This is a very long
tightly wound coil of wire carrying a current. In
cross section it can be approximated as two
current sheets with current flows in opposite
directions.
cross section view
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near uniform B-field inside, none outside
cut away side view
Now choose an appropriate Amperian loop.
evaluate the line integral
If there are n turns of wire per unit length,
each carrying current I then it is easy to
evaluate the total current through the loop
Apply Amperes law
UCSD Ph 2B lectures by George M. Fuller
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Mutual Inductance
ExampleConsider two coils, one encircling
another as shown in the cutaway drawing
Suppose that current I1 runs through coil 1 which
has N1 turns (wires) in length l and cross
sectional area A1. The B-field has magnitude
The magnetic flux through one loop of coil 1 is
through all of coil 2
If coil 2 has N2 turns in length l, the EMF in
coil 2 from a time varying current in coil 1 is
where we define the mutual inductance to be
We can write this result as
and we can show that
by similar reasoning we get
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Mutual Inductance
In fact, we can generalize this idea for any two
conducting circuits of arbitrary shape and
relative orientation. For example, for circuit 1
carrying current I1 and another circuit, circuit
2, the mutual inductance is the ratio of the
magnetic flux threaded by circuit 2 from the
magnetic field created by the current in circuit
1 to I1 . There will be an induced EMF in circuit
2 from a time-varying current in circuit 1 (here
we assume that any time-varying current varies
slowly with time). For example,
circuit 2
circuit 1
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Self Inductance
If there are currents flowing (and possibly
varying in time) in both circuits
simultaneously, then the magnetic flux through
either of the circuits individually will be the
sum of the separate magnetic fluxes from the
currents flowing in both circuits.
The induced EMF in one of the circuits therefore
will depend on the rate of change of current in
both circuits. For example, we can write the
induced EMF in circuit 2 as
where L is the self inductance of circuit 2.
Let us consider circuit 2 as completely isolated
(e.g., circuit 1 doesnt exist). Any circuit has
an interesting property its own magnetic field
(stemming from its own current) produces a
magnetic flux through the circuit which opposes
changes in the current. This is self inductance,
defined as the ratio of the magnetic flux through
the circuit to the current
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Dimensions of Inductance (both self and mutual
inductance)
henry (volt-second)/ampere
UCSD Ph 2B lectures by George M. Fuller
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Example Problem 11, Chapter 32 What is the self
inductance of a solenoid 50 cm long and 4.0 cm in
diameter that contains N 1000 turns of wire?
Solution the magnetic field in the solenoid is
taken to be uniform with a magnitude, as we have
seen before, that depends on the number of turns
per unit length n
In this case because the field is uniform and
normal to the plane of each turn, the magnetic
flux through each turn is just the product of
magnetic field magnitude and the cross sectional
area of the solenoid
The magnetic flux through the whole solenoid is
then
The self inductance is then
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For an isolated circuit with self inductance L,
Faradays law shows that any attempt to change
the current flowing in the circuit is countered
by a so-called back EMF given by
When the current is steady (zero rate of change)
there is no back EMF and the equivalent inductor
element in the circuit behaves like a piece of
wire. However, if the current does change with
time, then the inductor behaves like a
battery, delivering an EMF given by Faradays law
as above and directed to drive a current which
opposes the change in current.
An inductor element in a circuit behaves like a
battery as follows . . .
-

I

-
UCSD Ph 2B lectures by George M. Fuller
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Circuits with Inductors the RL Circuit Case 1
current build-up
Imagine that you have an open circuit as shown
and that at time t0 you close the circuit. What
happens?
R
Current I starts at zero and builds up with time.
loop law gives
EL -LdI/dt
E0
for a self inductance we have
differentiating our loop law equation w.r.t. time
we can integrate both sides of this to give
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So in this example the current as a function of
time must be (from Ohms law)
Note that this has the expected physical
behavior at time t t0 there is no
current flowing (we took this as a basis for
constructing this expression) and only for
times satisfying (R/L)(t - t0) gtgt 0 does the
inductor disappear and the current asymptote
to its final, steady value.
Now we have a quantitative measure of the
inertia brought to an electrical circuit by
inductance significant changes in the current
cannot occur on timescales much shorter than L/R.
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Circuits with Inductors the RL Circuit Case 2
current decay
Imagine that you have a circuit that has had
plenty of time to come to equilibrium. At time t0
you throw the switch allowing the current to die
away. What happens?
R
A steady current I0 is flowing when the switch is
thrown.
loop law then gives
EL -L dI/dt
E0
for a self inductance we have
substituting this in, the loop law equation
becomes
we can integrate both sides of this to give
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Short and long time behavior of an RL-circuit
R1
Close the switch
L
R2
R1
At very short times no current flows through the
inductor so it behaves like an open circuit and
all current flows through both resistors IE
/(R1R2).
R2
R1
At very long times the current has built up to a
steady state value in through the inductor and so
it behaves like a (zero resistance) wire. In
this case, the inductor element acts like a
short circuit and no current flows through R2 .
The current in the other resistor is I E /R1
R2
UCSD Ph 2B lectures by George M. Fuller
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Open the switch again and watch out!
The current I E/(R1) will instantaneously
disappear through resistor R1 but, of course, it
cannot change instantaneously through the
inductor. A current will continue to flow in the
cut-off circuit shown
R1
L
R2
What is the voltage across resistor R2 ?
The inductor will be fighting like mad to force a
current through R2. If there is nothing in
parallel with the inductor, this is like R2
infinity. In this case, watch out!
UCSD Ph 2B lectures by George M. Fuller
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Magnetic Energy
Consider our previous example of an RL-circuit
starting up from zero initial current. We found
in that case that the loop law gives
Now take this equation and multiply both sides by
the instantaneous current flowing
Remember that the EMF in the inductor is
Therefore
rate at which heat is generated by resistor
rate at which battery pumps energy into circuit
rate at which energy is extracted from circuit
and pumped into the inductor
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We just found that the rate at which the inductor
is storing up energy is
Therefore, in time interval dt the amount of
energy stored in the inductor is
If the current in the circuit is brought up from
zero to I then the energy stored in the
inductor must be
This energy is stored in the magnetic field in
the inductor element.
UCSD Ph 2B lectures by George M. Fuller
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Remember that the self inductance of a solenoid
with n turns per unit length, length l, and
cross sectional area A is
and the magnetic field is
Therefore, the magnetic energy stored in the
solenoid is
From this we can identify an energy per unit
volume associated with any magnetic field, as it
turns out
Likewise, we found earlier in studying
capacitors that there is an energy per unit
volume associated with electric fields
UCSD Ph 2B lectures by George M. Fuller