Moving charges – currents Ampere’s Law Gauss’ Law in magnetism Magnetic materials
s
dr̂
r̂
outdB
indB
I
P
P
rB
sB
ˆ
d
dd
sin
12
dB
dsdB
IdB
rdB
2
0ˆ
4 r
Idd
rsB
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permeability of free space
2
0ˆ
4 r
dI rsB
cosˆˆˆˆ dxrdsd kkrs
kkB ˆcos
4ˆ
2
0
r
dxIdBd
2
0 cos
4 r
dxIdB
2
2
cossec
tan
cos
addadx
ax
ar
d
a
I
a
daIdB cos
4cos
coscos
4
0
22
2
0
2100 sinsin
4cos
4
2
1
a
Id
a
IB
If the wire is very long,
90
90
2
1
then
a
IB
2
0
2
0 90sin
4 R
idsdB
0
0
4d
R
idBB
Rdds
R
iB
4
0
R
iB
2
0 Full Circle ( 2)
la
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a
IlIlBIF
22
120201211
BFFF 21
a
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l
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120
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
Irr
IdsBd 0
0 22
.
sB
Id 0. sB
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.
IrBdsBd 02. sB
r
IB
2
0
For r >= R
IR
rI
R
r
I
I
2
2
2
2
I
R
rIrBd
2
2
002. sB
For r < R
rR
IB
2
0
2
By symmetry, B is constant over
loop 1 and tangent to it
Bdsd sB.
NIrBdsBd 02. sB
r
NIB
2
0
Outside the toroid: 0B(Consider loop 2 whose plane is
perpendicular to the screen)
BldsBddpathpath
11
.. sBsB
NId 0. sB
nIB 0 where lNn
Along path 2 and 4, B is perpendicular to ds
Along path 3, B=0
Consider loop 2
AB dB .
cosBAB
For a uniform field making
an angle with the surface
normal:
(Weber=Wb=T.m2)
0B ABBB max,
Unlike electrical fields,
magnetic field lines end
on themselves, forming
loops. (no magnetic
monopoles)
Magnetic Field Lines Electric Field Lines
0. AB d
Aμ
I
R
IB
2
0
Current carrying loop Electron in orbit
2T
r
eve
T
eI
22
r
v
rev
rr
evIA
22
2
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ee m
enL
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22
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Electron spin TJm
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24
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.
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