Lecture 18
Magnetic Vector Potential,
Gauss Law of Magnetostatics,
Magnetic Flux
Sections: 7.5, 7.6, 7.7
Homework: See homework file
LECTURE 18
slide 1
Magnetic Vector Potential
• the magnetic flux density B can be represented as the curl of a
vector – the magnetic vector potential A
B   A
• this can be proven using the Biot-Savart law (optional)
B( P ) 

CQ
IdL
Q
aQP
RQP
aQP

I (Q ) dLQ  2 ,
4
RQP
P
 1 
aQP
  P 

2
RQP
 RQP 
LECTURE 18
slide 2
Magnetic Vector Potential – optional
 1 
aQP
 proving that 2   P 

RQP
 RQP 
R QP  ( xP  xQ )a x  ( yP  yQ )a y  ( z P  zQ )a z
RQP  ( xP  xQ ) 2  ( yP  yQ ) 2  ( z P  zQ ) 2
 1
P 
 RQP

RQP
RQP 
1  RQP
ax 
ay 
az 
 2 
RQP  xP
yP
z P


RQP 1 2( xP  xQ ) ( xP  xQ )
 

xP
2
RQP
RQP
 1 
aQP
 1 
 1 
 P 
side note:  P 
 Q 
 2


R
R
QP
 QP 
 RQP 
 RQP 
( xP  xQ )a x  ( yP  yQ )a y  ( z P  zQ )a z
aQP 
LECTURE
18
slide 3
RQP
Magnetic Vector Potential – optional
 1 

I (Q ) dLQ   P 
 B( P )   

4
 RQP 
C
Q
 use the vector identity F  V  V   F    (VF )
 1
  I (Q ) dLQ
 

I (Q ) dLQ    P  
 B( P )    
P  

  4
 4 RQP
CQ 
 RQP
0
  I ( Q ) dL Q 
 B( P )  
  P   4 RQP 
CQ
 curl operator does not depend on the point of integration Q
  I ( Q ) dL Q 
B( P )   P   

R

4
QP

CQ 

A( P)
LECTURE 18
slide 4


 
Magnetic Vector Potential in Terms of Current Element – optional
• magnetic vector potential due to line currents

A( P ) 
4

CQ
I Q dLQ
, Wb/m or
RQP
 I Q dLQ
dA Q ( P ) 

4 RQP
• dA is in the same direction as the current element generating it
– simpler than the Biot-Savart law for H
• analogous expressions are obtained for surface and volume
current distributions

A( P ) 
4
K (Q )
 RQP dsQ
SQ

A( P ) 
4
J (Q )
 RQP dvQ
vQ
• analogy with electric scalar potential V ( P ) 
1
4 
vQ
LECTURE 18
v (Q)
RQP
slide 5
dvQ
Gauss Law of Magnetism
B   A
 B  0
use identity    F  0
Gauss law of magnetism in differential form
• compare with Gauss law of electrostatics   D  v
• there are no magnetic charge monopoles
"CuttingABarMagnet" by Sbyrnes321 - Own work. Licensed
under CC0 via Wikimedia Commons http://commons.wikimedia.org/wiki/File:CuttingABarMagnet.s
vg#/media/File:CuttingABarMagnet.svg
• the elementary magnetic source is a dipole
LECTURE 18
slide 6
Gauss Law of Magnetism - 2
• use Gauss integral theorem to obtain Gauss Law of Magnetism
in integral form
   Bdv  0  
 B  ds
v


 B  ds  0
S[ v ]
Gauss law of magnetism in integral form
S
the magnetic flux through any closed surface is zero
• geometrical interpretation: field
lines are closed loops (no begin
and end points)
LECTURE 18
slide 7
Magnetic Flux
• definition
   B  ds, Wb
S
• Gauss law of magnetism (flux through a closed surface is zero)

 B  ds  0
S
• flux through an open surface and A
   B  ds   (  A )  ds
S
S
 
Stokes’ theorem
 A  dL
C[ S ]
• flux is essential in defining inductance and induced EMF
LECTURE 18
slide 8
You have learned:
B is the curl of the magnetic vector potential A B   H    A
the net magnetic flux Φ over a closed surface is always zero, i.e.,
it appears that there are no magnetic charges

 B  ds  0
S
A depends on the current density J in a manner analogous to the
way V depends on ρv

A( P) 
4
J (Q )
 RQP dvQ
vQ
compare
V ( P) 
1
4 
vQ
LECTURE 18
v (Q)
RQP
slide 9
dvQ