Transcript Lecture #21 04/14/05

Announcements
• Office hrs 2-3 pm
– If you cannot make my office hrs, email for
appointment!
• Quiz IV
–
–
–
–
–
9 MC, 1 essay and 3 problems
April 21st
Ch 29.6-10,30,31, 33.1-5
Reading quizzes 15-19
HW 16-20
V  E cos t 
E
C
–
d 2  V 
V  CL
2
dt
+
LC - Circuits
L
S1
S2
  1/ LC
dQ
d
I
 C  V   C sin  t 
dt
dt
These equations
LC – Circuits and
Harmonic Oscillators
V  CL
d
2
 V 
dt
2
V  E cos t 
 m d 2x
x
2
k dt
x  A cos(t )
There are many correspondances between
These equations
electrical and mechanical systems!
LC – Circuits and Energy
+
V  E cos t 
–
I  CE sin t 
C
L
S1
  1/ LC
S2
At an arbitrary time t, where is the energy stored in this circuit?
A) In the capacitor
B) In the inductor
C) Alternately in the capacitor or the inductor
D) What energy?
U C  C  V   CE cos t 
2
1
2
1
2
2
2
U tot  12 CE 2
U L  LI  12 LC 2E 2 2 sin 2 t   12 CE 2 sin 2 t 
1
2
2
Reminder: Multiloops and dipoles
•If there are N loops basically all
identical, multiply by N
U   INA  B   INA  B
•All the properties of the wire are summarized in the
magnetic dipole moment of the wire
•Units of Am2 = J/T
  INA
U    B
  B
Magnetism in Matter: Orbital
•Magnetism at a macroscopic level arises from magnetism at a
microscopic level.
electron
If we consider a classical model of
an electron moving in a loop around
a nucleus, then we have a current
loop.
nuclei
 e
  
 2me

 L

Therefore we have a magnetic dipole
associated with the orbital motion
of the electron.
In most materials – but not all! – the
electron’s orbital magnetic dipoles cancel
each other out.
Magnetism in Matter: Spin
•Electrons (and other particles) have a second source of
angular momentum, spin.
This does not come from a spinning motion
about an axis!
The magnetic dipole moment due to spin is of similar magnitude to
the orbital magnetic moment.
Atoms with an odd number of electrons cannot have cancellation of
spin.
 e
  
 me

 S

The total magnetic dipole moment is a
vector sum of the spin and orbital moments.
Magnetism in Matter: Atomic
•At the atomic and molecular level, angular momentum, and
hence magnetic moments are quantized (take Phy 141 for
details)
For an electron, spin up or spin down
Orbital momentum can depends on the “electron shell”
Magnetism in Matter: Types
•Depending on details of the electronic structure on can have
permanent or induced magnetic moments – similar to the case
with dipoles.
Diamagnetic substances: induced magnetic moments counter to the
applied field.
All substances have a diamagnetic response, but it is weak.
Paramagnetic substances: the molecules have permanent moments, but are
weakly coupled and require a field to line them up, and the net moments
are in the direction of the field.
Ferromagnetic substances: the atoms have permanent moments and are
strongly coupled. The atoms can remain aligned in the absence of a field.
Gauss’s Law
•Magnetic field lines are always loops, never start or end on
anything (no magnetic monopoles)
•Net flux in or out of a region is zero
 B  dA  0
•Gauss’s Law for electic fields
 E   E  dA  4 ke qin
Four closed surfaces are shown. The areas Atop and Abot of
the top and bottom faces and the magnitudes Btop and
Bbot of the uniform magnetic fields through the top and
bottom faces are given. The fields are perpendicular to the
faces and are either inward or outward. Which surface has
the largest magnitude of flux through the curved faces?
Ampere’s Law
 B  ds   I
I
0
•Ampere’s law says that if we take the dot product of the field
and the length element and sum up (i.e. integrate) over a
closed loop, the result is proportional to the current through
the surface
•This is not quite the same as gauss’s law
A Problem with Ampere’s Law
•Consider a parallel plate capacitor that is
being charged
•Try Ampere’s Law on two nearly
identical surfaces/loops
I
 B  ds   I
 B  ds   I
1
0 1
0
2
0 2
 0 I  0
•Magnetic fields cannot be different just because
surfaces are chosen slightly different
•When current flows into a region, but not out,
Ampere’s law must be modified
I
Ampere’s Law Generalized
•When there is a net current flowing into a region, the charge
in the region must be changing, as must the electric field.
•By Gauss’s Law, the electric flux must be changing as well
•Change in electric flux creates magnetic fields, just like currents do
•Displacement current – proportional to time derivative of electric
flux
dE
Id  0
dt
dE
 B  ds  0 I  0 0 dt
Magnetic fields are created both
by currents carried by conductors
(conduction currents) and by
time-varying electric fields.
Generalized Ampere’s Law Tested
•Consider a parallel plate capacitor that is
being charged
•Try Ampere’s modified Law on two
nearly identical surfaces/loops
I
d  E1
 B1  ds  0 I1  0 0 dt
d Q
 0 0    0 I
dt   0 
dE2
 B2  ds  0 I 2  0 0 dt  0 I
I