Lecture 11 Magnetism of Matter: Maxwell

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Lecture 11 Magnetism of Matter Maxwells
Equations Chp. 32Wednesday Morning

• Cartoon -. Opening Demo -
• Warm-up problem
• Physlet
• Topics
• Finish up Mutual inductance
• Ferromagnetism
• Maxwell equations
• Displacement current
• Exam
• Demos

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What is Mutual Inductance? M
When two circuits are near one another and both
have currents changing, they can induce emfs in
each other.
On circuit boards you have to be careful you do
not put circuits near each other that have large
mutual inductance. They have to be oriented
carefully and even shielded.
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71. Two coils, connected as shown, separately
have inductances L1 and L2. Their mutual
inductance is M. (a) Show that this combination
can be replaced by a single coil of equivalent
inductance given by
We assume that the current is changing at
(nonzero) rate di/dt and calculate the total emf
across both coils.
First consider coil 1. The magnetic field due to
the current in that coil points to the left.
The magnetic field due to current in coil 2 also
points to the left. When the current increases,
both fields increase and both changes in flux
contribute emfs in the same direction.
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e
Thus, the induced emfs are
Therefore, the total emf across both coils is
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(b) How could the coils in this figure be
reconnected to yield an equivalent inductance of
We imagine reversing the leads of coil 2 so the
current enter at the back of the coil rather than
front (as pictured in the diagram). Then the
field produced by coil 2 at the site of coil 1 is
opposite to the field produced by coil 1 itself.
e
The fluxes have opposite signs. An increasing
current in coil 1 tends to increase the flux in
that coil, but an increasing current in coil 2
tends to decrease it.
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The emf across coil 1 is
Similarly, the emf across coil 2 is
The total emf across both coils is
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75. A rectangular loop of N closely packed turns
is positioned near a long, straight wire as shown
in the figure. (a) What is the mutual inductance
M for the loop-wire combination? (b) Evaluate M
for N 100, a 1.0 cm, b 8.0 cm, and l 30
cm.
(a) The flux over the loop cross section due to
the current i in the wire is given by
Thus,
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(b) Evaluate M for N 100, a 1.0 cm, b 8.0
cm, and l 30 cm.
(b) From the formula for M obtained,
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Ferromagnetism
Iron, cobalt, nickel, and rare earth alloys
exhibit ferromagnetism. The so called exchange
coupling causes electron magnetic moments of one
atom to align with electrons of other atoms. This
alignment produces magnetism. Whole groups of
atoms align and form domains. (See Figure 32-12
on page 756) A material becomes a magnet when
the domains line up adding all the magnetic
moments.You can actually hear the domains
shifting by bringing up an magnet and hear the
induced currents in the coil. Barkhausen Effect
Two other types of magnetic behavior are
paramagnetism or diamagnetism.
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What is the atomic origin of magnetism?
Electron spinning on its axis
Electron orbiting around the nucleus
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Spin Magnetic Dipole Moment of the Electron
S is the angular momentum due to the electrons
spin. It has units kg.m2/s. m has units of A.m2
- current times area Recall for a current loop,
the magnetic dipole moment current times area
of loop
In the quantum field theory of the electron, S
can not be measured. Only its component along
the z axis can be measured. In quantum physics,
there are only two values of the z component of
the electron spin.
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Therefore, only the z component of m can be
measured.Its two possible values are
Corresponding to the two values of the electron
spin quantum number 1/2 and -1/2
The above quantity is called the Bohr magneton
and is equal to
The magnetic moment of the electron is the prime
origin of ferromagnetism in materials.
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22. The dipole moment associated with an atom of
iron in an iron bar is 2.1x10-23 J/T. Assume that
all the atoms in the bar, which is 5.0 cm long
and has a cross-sectional area of 1.0 cm2, have
their dipole moments aligned. (a) What is the
dipole moment of the bar? (b) What torque must
be exerted to hold this magnet perpendicular to
an external field of 1.5 T? (The density of iron
is 7.9 g/cm3)
(a) The number of iron atoms in the iron bar is
Thus, the dipole moment of the bar is
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(C) Use the dipole formula to find the magnitude
and direction of the magnetic field 1cm from the
end of the bar magnet on its central axis at P.
5 cm
.
P
z
A 1 cm2
m 8.9 A.m2
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BigBite is a 50 ton electromagnet with a 25 cm by
100 cm gap
B 1 Tesla
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Maxwells EquationsIn 1873 he wrote down 4
equations which govern all classical
electromagnetic phenomena.
You already know two of them.
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A magnetic field changing with time can produce
an electric field Faradays law
Line integral of the electric field around the
wire equals the change of Magnetic flux through
the area Bounded by the loop
Electric lines curl around changing magnetic
field lines
Example
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Faradays Law
B is increasing in magnitude
Note that induced E field is in such a
direction that the B field it produces opposes
the original B field.
Note there is no electric potential associated
with the electric field induced by Faradays Law
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Can a changing electric field with time produce
an magnetic field
Yes it can and it is called Maxwells law of
induction
.
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Maxwells law of induction
Consider the charging of our circular plate
capacitor
B field also induced at point 2.
When capacitor stops charging B field disappears.
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Find the expression for the induced magnetic
field B that circulates around the electric field
lines of a charging circular parallel plate
capacitor
r lt R
Flux within the loop of radius r
0
E
B
r
R
rlt R
r gt R
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Ampere-Maxwells Law
Maxwell combined the above two equations to form
one equation
This term has units of current
How do we interpret this equation?
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What is the displacement current?
This is called the displacement current id
The term is really is a transfer of electric and
magnetic energy from one plate to the other
while the plates are being charged or discharged.
When charging stops, this term goes to zero.
Note it is time dependent.
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Show that the displacement current in the gap of
the two capacitor plates is equal to the real
current outside the gap
Can I detect the magnetic field associated with
displacement current?
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Calculation of id
First find the real current i
For the field inside a parallel plate capacitor
Solving for q
This is the real current i charging the capacitor.
Next find the displacement current
displacement current real current. No charge
actually moves across the gap.
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Calculate Magnetic field due to displacement
current
Current is uniformly spread over the circular
plates of the capacitor. Imagine it to be just a
large wire of diameter R. Then use the formula
for the magnetic field inside a wire.
Inside the capacitor
Outside the capacitor
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Question 11 A circular capacitor of radius R is
being charged through a wire of radius R0. Which
of the points a, b, c, and d correspond to points
1, 2, and 3 on the graph
Where is the radius R0 and R on the graph?
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(a) What is the displacement current id through
the region between the plates?
At any instant the displacement current id in the
gap between the plates equals the conduction
current i in the wires. Thus, id i 3.0 A.
(b) What is dE/dt in this region?
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(c) What is the displacement current through the
square, dashed path between the plates?
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(e) What is the value of B on this path?????
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Summary of Maxwell EquationsIntegral form
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Warm up set 10 Due 800 am Tuesday

• HRW6 31.TB.02. 120186 Suppose this page is
  perpendicular to a uniform magnetic field and the
• magnetic flux through it is 5 Wb. If the page is
  turned by 30 around an edge the flux
• through it will be
• 4.3 Wb
• 10 Wb
• 5.8 Wb
• 2.5 Wb
• 5 Wb
• 2. HRW6 31.TB.08. 120192 Faraday's law states
  that an induced emf is proportional to
• the rate of change of the electric field
• the rate of change of the magnetic field
• zero
• the rate of change of the magnetic flux
• the rate of change of the electric flux
• 3. HRW6 31.TB.09. 120193 The emf that appears
  in Faraday's law is
• around a conducting circuit