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Lecture 9
Vector Magnetic Potential
Biot Savart Law
Prof. Viviana Vladutescu
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Figure 1: The magnetic (H-field)
streamlines inside and outside a
single thick wire.
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Figure 2: The H-field magnitude
inside and outside the thick wire
with uniform current density
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Figure 3: The H-field magnitude
inside and outside the thick
conductors of a coaxial line.
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Figure 1: The vector potential in
the cross-section of a wire with
uniform current distribution.
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Figure 2: Comparison between the magnetic vector potential
component of a wire with uniformly distributed current and the
electric potential Vof the equivalent cylinder with uniformly
distributed charge.
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JAA
AAAAA
JA
0
2
20
)(
)()()(
JAA 020 Vector Poissons equation
Laplacian Operator (Divergence of a gradient)
Poissons Equation
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In electrostatics
ED
VE
E
D
0
V
EE
V2 Poissons Equationin electrostatics
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4
4
1
00
2
00
2
dv
R
JAJA
dvR
VV
v
v
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Magnetic Flux
(Wb))(
cs
s
ldAdsA
dsB
The line integral of the vector magnetic potential A around
any closed path equals the total magnetic flux passing
through area enclosed by the path
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The Biot-Savart Law relates magnetic fields to the currents
which are their sources. In a similar manner, CoulombsLaw
relates electric fields to the point charges which are their
sources. Finding the magnetic field resulting from a currentdistribution involves the vector product, and is inherently a
calculus problem when the distance from the current to the
field point is continuously changing.
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)( TAB
40
c RldIA
4
0
c R
ldIB
GfGfGf
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1140
c
ldR
ldR
IB
211R
aR
R
By using
(T)4 20
c
R
RaldIB
(see eq 6.31)
Biot-Savart Law
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2
0
4
R
aldIBd
BdB
R
c
In two steps
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Illustration of the law of BiotSavart showing
magnetic field arising from a differential segment of
current.
2
12
12112
4 R
aLdIHd
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rzR arazaR
Example1
Component values for the equation to find the
magnetic field intensity resulting from an infinite
length line of current on the z-axis. (ex 6-4)
r
aIH
rzr
zaIr
rz
dzaIr
rz
arazaIdzH rzz
24
)(4)(4
)(
222
23
2223
22
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Example 2
We want to find Hat height habove
a ring of current centered in the xyplane.
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The component values shown for use in the BiotSavart
equation.
2
02
322 )(4
)(
ah
aaahaIad
H rz
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The radial components of Hcancel
by symmetry.
2
322
2
2
02
322
2
2
4
ah
aIaH
dah
aIaH
z
z
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Solenoid
Many turns of insulated wire coiled in the shape of a cylinder.
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For a set N number of loops around a ferrite
core, the flux generated is the same even when
the loops are bunched together.
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Example : A simple toroid wrapped with N turns modeled
by a magnetic circuit. Determine B inside the closely wound
toroidal coil.
b
a
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)()(,2
2
0
0
abrabr
NIaaBB
NIrBldB
Amperes Law
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a) An iron bar attached to an electromagnet.b) The bar displaced by a differential length d.
Electromagnets
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Levitated trains: Maglev prototype
Electromagnet supporting a
bar of mass m.
Applications
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Wilhelm Weber (1804-1891). Electromagnetism.