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
 A   0 J  A 
4
2
J
dv
v R
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



R
4 c
 
  f G  f  G  f  G
Biot-Savart Law

0 I  1
 1
B


d
l



d
l





4 c  R
 R

1
1
By using     a R 2 (see eq 6.31)
R
R
0 I d l  aR
B
2

4 c R
(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
4R12
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 a R   z a z  r ar

H


Idz a z  ( z a z  r ar )
4 ( z  r )
2
2
3

2

Ir a
4

 (z

dz
2
r )
Ir a 
I a

z

 2 2 2 H 
4  r z  r  
2r
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
2 h a
2
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  2rB   NI
0
0 NI
 B  B a  a
, (b  a)  r  (b  a)
2r
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.