



Moving charges – currents
Ampere’s Law
Gauss’ Law in magnetism
Magnetic materials
P
r̂


ds
r̂
P

dB in
 0 Ids  rˆ
dB 
4 r 2

dB out
I


dB  d s

dB  rˆ
1
dB  2
r
dB  I
dB  ds
dB  sin 
 0 I ds  rˆ
B
4  r 2
0  4 107 T .m / A
permeability of free space

ds  rˆ  kˆ ds  rˆ  kˆ dx cos 

 I dx cos  ˆ
dB  dB kˆ  0
k
2
4
r
dB 
0 I dx cos 
4
r2
a
cos 
x  a tan 
0 I a cos 2  cos d 0 I
dB 

cos d
4
a 2 cos 2 
4a
ad
dx  a sec d  
cos 2 
I
I
B  0  cos d  0 sin 1  sin  2 
4a 
4a
r
2
2
1
If the wire is very long,
1  90
 2  90
then
0 I
B
2a
dB 
0 ids sin 90
4
R2
ds  Rd 

 0i
B   dB 
d 

4R 0
B
B
0i
4R
 0i
2R
Full Circle (  2)
 I   I I
F1  I1lB2  I1l  0 2   0 2 1 l
2a
 2a 
F1  F2  FB
FB 0 I 2 I1

l
2a
On a computer chip, two conducting strips carry charge
from P to Q and from R to S. If the current direction is
reversed in both wires, the net magnetic force of strip 1 on
strip 2
1. remains the same.
2. reverses.
3. changes in magnitude, but
not in direction.
4. changes to some other
direction.
5. other
0 I
 B.ds  B ds  2r 2r   0 I
 
 B.ds  0 I
A line integral of B.ds
around a closed path
equals 0I, where I is
the total continuous
current passing
through any surface
bounded by the
closed path.
For r >= R
 
 B.ds  B ds  B2r   0 I
B
0 I
2r
For r < R
I  r 2
 2
I R
r2
I  2 I
R
 
 r2
 B.ds  B2r   0 I   0  R 2
 I 
B   0 2 r
 2R 

I 

By symmetry, B is constant over
loop 1 and tangent to it
B.ds  Bds
 B.ds  B ds  B2r    NI
0
0 NI
B
2r
Outside the toroid:
B0
(Consider loop 2 whose plane is
perpendicular to the screen)
Consider loop 2
Along path 2 and 4, B is perpendicular to ds
Along path 3, B=0
 B.ds   B.ds  B  ds  Bl
path1
path1
 B.ds   NI
0
B  0 nI
where
nN
l
 B   B.dA
(Weber=Wb=T.m2)
For a uniform field making
an angle  with the surface
normal:
 B  BA cos 
B  0
 B   B,max  AB
Unlike electrical fields,
magnetic field lines end
on themselves, forming
loops. (no magnetic
monopoles)
 B.dA  0
Electric Field Lines
Magnetic Field Lines
Electron in orbit
Current carrying loop
B
T
0 I
2

2R
I


μ  IA

v
r
L  me vr
e e ev


T 2 2r
  IA 
ev 2 ev
r  r
2r
2
 e 
 e 
 L  n 2 

 2me 
 2me 
  
Electron spin
 e 
  9.27 1024 J / T
 2me 
 spin  


The net orbital magnetic moment in most
substances is zero or very small since the
moments of electrons in orbit cancel each
other out.
Similarly, the electron spin does not
contribute a large magnetic moment in
substances with an even number of electrons
because electrons pair up with ones of
opposite spin.
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

In certain crystals, neighboring groups of atoms called domains have
their magnetic moments aligned.
A ferromagnetic crystal can be given a permanent magnetization by
applying an external field.
Examples include Fe, Co, Ni.
Unmagnetized crystal
has domains of aligned
magnetic moments.
An external field
increases the sizes
of the domains
aligned with it.
As the external field
gets larger, the
unaligned domains
become smaller.



Paramagnetic substances have a small but
positive magnetism that comes from atoms
with permanent magnetic moments.
An external field can align these moments for
a net magnetization but the effect is weak
and not permanent.
Certain organic compounds such as
myoglobin are paramagnetic.
A very small effect
caused by the induction
of an opposing field in
the atoms by an
external field.
 Superconductors
exhibit perfect
diamagnetism
(Meissner effect).

Biot-Savart Law gives the magnetic field due to a
current carrying wire.
 Ampere’s Law can simplify these calculations for
cases of high symmetry.
 The magnetic flux through a closed surface is zero.
 Solenoids and toroids can confine magnetic fields.
 Orbital and spin magnetic moments of electrons give
rise to magnetism in matter.


Reading Assignment
 Chapter 31 – Faraday’s Law

WebAssign: Assignment 8