Lecture 9 Vector Magnetic Potential Biot Savart Law
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Transcript Lecture 9 Vector Magnetic Potential Biot Savart Law
Lecture 9
Vector Magnetic Potential
Biot Savart Law
Prof. Viviana Vladutescu
Figure 1: The magnetic (H-field)
streamlines inside and outside a
single thick wire.
Figure 2: The H-field magnitude
inside and outside the thick wire
with uniform current density
Figure 3: The H-field magnitude
inside and outside the thick
conductors of a coaxial line.
Vector Magnetic Potential
B 0
A 0
B A (T )
A - vector magnetic potential (Wb/m)
Figure 1: The vector potential in
the cross-section of a wire with
uniform current distribution.
Figure 2: Comparison between the magnetic vector potential
component of a wire with uniformly distributed current and the
electric potential V of the equivalent cylinder with uniformly
distributed charge.
Poisson’s Equation
A 0 J
A ( A) ( ) A ( A) A
2
Laplacian Operator (Divergence of a gradient)
( A) A 0 J
2
2
A 0 A 0 J
Vector Poisson’s equation
D
In electrostatics
E 0
E V
D E
E E
V
V
2
Poisson’s Equation
in electrostatics
1
V V
dv
0
4 0 v R
2
0 J
A 0 J A
dv
4 v R
2
Magnetic Flux
B ds
s
( A) ds A d l (Wb)
s
c
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
Biot Savart Law and
Applications
The Biot-Savart Law relates magnetic fields to the currents
which are their sources. In a similar manner, Coulomb’s Law
relates electric fields to the point charges which are their
sources. Finding the magnetic field resulting from a current
distribution involves the vector product, and is inherently a
calculus problem when the distance from the current to the
field point is continuously changing.
B A (T )
0 I d l
A
4 c R
dl
0 I
B
4 c
R
f G f G f G
Biot-Savart Law
0 I 1
1
B
d
l
d
l
4 c R
R
By using
1
1
aR 2
R
R
0 I d l aR
B
2
4 c R
(see eq 6.31)
(T )
In two steps
B dB
c
0 I d l aR
dB
2
4 R
Illustration of the law of Biot–Savart showing
magnetic field arising from a differential segment of
current.
I1d L1 a12
dH2
2
4R12
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)
RaR z az r ar
H
Idza z ( z a z r ar )
4 ( z r )
2
2
3
2
Ira
4
(z
dz
2
r )
Ira
I a
z
2 2 2 H
4 r z r
2r
2
3
2
Example 2
We want to find H at height h above
a ring of current centered in the x –
y plane.
2
H
0
Iad a (ha z a ar )
4 (h a )
2
2
3
2
The component values shown for use in the Biot–Savart
equation.
The radial components of H cancel
by symmetry.
H
2
2
Ia a z
4 h a
H
2
2
3
2
d
0
2
Ia a z
2h
2
a
2
3
2
Solenoid
Many turns of insulated wire coiled in the shape of a cylinder.
For a set N number of loops around a ferrite
core, the flux generated is the same even when
the loops are bunched together.
Example : A simple toroid wrapped with N turns modeled by
a magnetic circuit. Determine B inside the closely wound
toroidal coil.
a
b
Ampere’s Law
B d l 2rB NI
0
0 NI
B B a a
, (b a ) r (b a )
2r
Electromagnets
a) An iron bar attached to an electromagnet.
b) The bar displaced by a differential length d.
Applications
Levitated trains: Maglev prototype
Electromagnet supporting a
bar of mass m.
Wilhelm Weber (1804-1891). Electromagnetism.