sources of the magnetic field - wake forest...
TRANSCRIPT
Sources of the Magnetic FieldMoving charges – currentsAmpere’s Law Gauss’ Law in magnetismMagnetic materials
Biot-Savart
Law
sr
dr̂
r̂
outdBr
indBr
I
P′
P
θ
rB
sBˆ⊥
⊥r
rr
d
dd
θsin
12
∝∝∝
∝
dBdsdBIdBr
dB
20 ˆ
4 rIdd rsB ×
=π
μr AmT /.104 70
−×= πμpermeability of free space∫
×= 2
0 ˆ4 r
dI rsBπ
μr
Magnetic Field Surrounding a Thin, Straight Conductor
θcosˆˆˆˆ dxrdsd kkrs =×=×r
( ) kkB ˆcos4
ˆ2
0
rdxIdBd θ
πμ
==r
20 cos
4 rdxIdB θ
πμ
=
θθθθ
θθ
22
cossec
tancos
addadx
ax
ar
−=−=
−=
=θθ
πμ
θθθθ
πμ d
aI
adaIdB cos
4coscoscos
40
22
20 ==
( )2100 sinsin
4cos
4
2
1
θθπ
μθθπ
μ θ
θ
−== ∫ aId
aIB
If the wire is very long,°≈°−≈
9090
2
1
θθ then
aIB
πμ2
0=
Field Due to a Circular Arc of Wire
20 90sin
4 RidsdB °
=π
μ
∫∫ ′==φ
φπμ
0
0
4d
RidBB
φ′= Rdds
RiB
πφμ
40=
RiB
20μ
= Full Circle (φ = 2π)
Magnetic Force Between Two Parallel Conductors
laII
aIlIlBIF
πμ
πμ
2212020
1211 =⎟⎠⎞
⎜⎝⎛==
BFFF == 21
aII
lFB
πμ2
120=
Concept QuestionOn 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
Ampère’s
Law
( ) IrrIdsBd 0
0 22
. μππ
μ=== ∫∫ sB
Id 0. μ=∫ sB rr
A line integral of B.ds around a closed path equals μ0 I, where I is the total continuous current passing through any surface bounded by the closed path.
Long Current Carrying Wire
( ) IrBdsBd 02. μπ === ∫∫ sBrr
rIB
πμ2
0=
For r >= R
IRrI
Rr
II
2
2
2
2
=′
=′
ππ
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛=′==∫ I
RrIrBd 2
2
002. μμπsBrr
For r < R
rRIB ⎟
⎠⎞
⎜⎝⎛= 2
0
2πμ
The ToroidBy symmetry, B is constant over loop 1 and tangent to it
Bdsd =sB.
( ) NIrBdsBd 02. μπ === ∫∫ sB
rNIBπ
μ20=
Outside the toroid: 0≈B (Consider loop 2 whose plane is perpendicular to the screen)
The Solenoid
The Magnetic Field of a Solenoid
BldsBddpathpath
=== ∫∫∫11
.. sBsB
NId 0. μ=∫ sB
nIB 0μ= where lNn =
Along path 2 and 4, B is perpendicular to ds
Along path 3, B=0
Consider loop 2
Magnetic Flux
∫=Φ AB dB .
θcosBAB =Φ
For a uniform field making an angle θ with the surface normal:
(Weber=Wb=T.m2)
0=ΦB ABBB =Φ=Φ max,
Gauss’
Law in Magnetism
Unlike electrical fields, magnetic field lines end on themselves, forming loops. (no magnetic monopoles)
Magnetic Field LinesElectric Field Lines
0. =∫ AB d
Magnetic Moments of Atoms
Aμrr I=
RIB
20μ
=
Current carrying loop Electron in orbit
ωπ2
=T
reve
TeI
ππω
22===
rv
=ω
revrr
evIA22
2 === ππ
μ
vrmL e=
h⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
ee menL
me
22
2μ
Electron spin TJme
espin /1027.9
224−×=⎟⎟
⎠
⎞⎜⎜⎝
⎛= hμ
Magnetic Moments of AtomsThe 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.
FerromagnetismIn 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.
ParamagnetismParamagnetic 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.
DiamagnetismA very small effect caused by the induction of an opposing field in the atoms by an external field.Superconductors exhibit perfect diamagnetism (Meissner effect).
SummaryBiot-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.
For Next ClassReading Assignment
Chapter 31 – Faraday’s Law
WebAssign: Assignment 8