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4). Ampere’s Law and Applications. As far as possible, by analogy with Electrostatics B is “magnetic flux density” or “magnetic induction” Units: weber per square metre (Wb m -2 ) or tesla (T) Magnetostatics in vacuum, then magnetic media based on “magnetic dipole moment”. I. - PowerPoint PPT Presentation
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4). Ampere’s Law and Applications
• As far as possible, by analogy with Electrostatics
• B is “magnetic flux density” or “magnetic induction”
• Units: weber per square metre (Wbm-2) or tesla (T)
• Magnetostatics in vacuum, then magnetic mediabased on “magnetic dipole moment”
Biot-Savart Law• The analogue of Coulomb’s Law is
the Biot-Savart Law
• Consider a current loop (I)
• For element dℓ there is an associated element field dB
dB perpendicular to both dℓ’ and r-r’same 1/(4r2) dependence o is “permeability of free space” defined as 4 x 10-7 Wb A-1 m-1
Integrate to get B-S Law
O
r
r’
r-r’
dB(r)
dℓ’
3o )x('d
4)(d
r'r
r'rrB
3
o )x('d
4)(
r'r
r'rrB
B-S Law examples
(1) Infinitely long straight conductor
dℓ and r, r’ in the page dB is out of the page
B forms circlescentred on the conductor
Apply B-S Law to get:
r
I
2B o
r - r’
dB
I
dℓ
B
r’
Or
sin = cos 1/222 zr
r
z
B-S Law examples(2) “on-axis” field of circular loop
Loop perpendicular to page, radius a
dℓ out of page and r, r’ in the page On-axis element dB is in the page, perpendicular to r - r’, at to axis.
Magnitude of element dB
Integrating around loop, only z-components of dB surviveThe on-axis field is “axial”
1/2222o
z2o
za
a
'-
acoscos
'-
d
4dB
'-
d
4dB
rrrrrr
II
r - r’ dB
I
dℓ
zdBz
r’r
a
On-axis field of circular loop
Introduce axial distance z, where |r-r’|2 = a2 + z2
2 limiting cases:
3
2o
2o
2o
zaxison
'-2
aa2cos
'-4
dcos'-4
dBB
rrrr
rr
I
I
I
23
22
2o
axisonza2
aB
I
3
2oaz
axisono0z
axison 2z
aBand
2aB
I
I
r - r’ dB
I
dℓ
zdBz
r’r
a
Magnetic dipole momentThe off-axis field of circular loop ismuch more complex. For z >> a it is identical to that of the electric dipole
m “current times area” vs p “charge times distance”
loop currentby enclosed area
z a oramwhere
sincos 2r4
m
sin cos 2r4
p
22
3o
3o
ˆI I I
ˆ ˆ
ˆˆ
m
rB
rE
rm
B field of large current loop• Electrostatics – began with sheet of electric monopoles• Magnetostatics – begin sheet of magnetic dipoles• Sheet of magnetic dipoles equivalent to current loop• Magnetic moment for one dipole m = I area
for loop M = I A area A
• Magnetic dipoles one current loop• Evaluate B field along axis passing through loop
B field of large current loop• Consider line integral B.dℓ from loop• Contour C is closed by large semi-circle which contributes
zero to line integral
2za
dza
circle)(semi 0za
dza
2.d
3/222
2
o3/222
2o
II
C
B
.dB
oI/2
oI
z→+∞z→-∞
I (enclosed by C)
Ca
Electrostatic potential of dipole sheet
• Now consider line integral E.dℓ from sheet of electric dipoles• m = I I = m/ (density of magnetic moments)
• Replace I by Np (dipole moment density) and o by 1/o
• Contour C is again closed by large semi-circle which contributes zero to line integral
.dE
Np/2o
-Np/2o
0
C
.dcircle)(semi 0.d EE
Electric magnetic
Field reverses no reversal
Differential form of Ampere’s Law
S
SjB .d.d oenclo I
S
j
B
dℓ
Sj.dd I
Obtain enclosed current as integral of current density
Apply Stokes’ theorem
Integration surface is arbitrary
Must be true point wise
SS
SjSBB .dd.d o.
jB o
Ampere’s Law examples(1) Infinitely long, thin conductor
B is azimuthal, constant on circle of radius r
Exercise: find radial profile of B inside and outside conductor of radius R
r2Br 2 B.d o
oenclo
I II
B
B
r 2B
R 2
rB
oRr
2o
Rr
I
I
B
rR
SolenoidDistributed-coiled conductorKey parameter: n loops/metre
If finite length, sum individual loops via B-S Law
If infinite length, apply Ampere’s Law B constant and axial inside, zero outsideRectangular path, axial length L
(use label Bvac to distinguish from core-filled solenoids)
solenoid is to magnetostatics what capacitor is to electrostatics
I II nBnLLB.d ovacovacenclovac B
IL
B
I
Relative permeability
Recall how field in vacuum capacitor is reduced when dielectric medium is inserted; always reduction, whether medium is polar or non-polar:
is the analogous expression
when magnetic medium is inserted in the vacuum solenoid.
Complication: the B field can be reduced or increased, depending on the type of magnetic medium
vacrr
vac BBE
E
Magnetic vector potential
0 x x
-.d
0
E
EE For an electrostatic field
We cannot therefore represent B by e.g. the gradient of a scalarsince
Magnetostatic field, try
B is unchanged by
)
o
o
.( always 0 . also
zero) not (x
EB
jB rhs
later) (see xx x
0 x
x
AB
AB
AB
..
0xx'x
AAA
AA
'
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