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1

To begin our study of magnetostatics with
Ampere’s law of force; magnetic flux density;
Lorentz force; Biot-Savart law; applications of
Ampere’s law in integral form; vector
magnetic potential; magnetic dipole; and
magnetic flux.
Lecture 7
2
Fundamental laws of
classical electromagnetics
Special
cases
Electrostatics
Statics:
Input from
other
disciplines
Maxwell’s
equations
Magnetostatics

0
t
Geometric
Optics
Electromagnetic
waves
Transmission
Line
Theory
Circuit
Theory
Kirchoff’s
Laws
d  
Lecture 7
3



Magnetostatics is the branch of
electromagnetics dealing with the effects
of electric charges in steady motion (i.e,
steady current or DC).
The fundamental law of magnetostatics is
Ampere’s law of force.
Ampere’s law of force is analogous to
Coulomb’s law in electrostatics.
Lecture 7
4

In magnetostatics, the magnetic field is
produced by steady currents. The
magnetostatic field does not allow for
◦ inductive coupling between circuits
◦ coupling between electric and magnetic fields
Lecture 7
5
Ampere’s law of force is the “law of
action” between current carrying
circuits.
 Ampere’s law of force gives the magnetic
force between two current carrying
circuits in an otherwise empty universe.
 Ampere’s law of force involves
complete circuits since current must
flow in closed loops.

Lecture 7
6

Experimental
facts:
◦ Two parallel wires
carrying current in
the same direction
attract.
◦ Two parallel wires
carrying current in
the opposite
directions repel.
F21 F12


I1
F21
I2
F12


I1
I2
Lecture 7
7

Experimental
facts:
◦ A short currentcarrying wire
oriented
perpendicular to a
long currentcarrying wire
experiences no
force.
F12 = 0

I2
I1
Lecture 7
8

Experimental facts:
◦ The magnitude of the force is inversely proportional
to the distance squared.
◦ The magnitude of the force is proportional to the
product of the currents carried by the two wires.
Lecture 7
9

The direction of the force established
by the experimental facts can be
mathematically represented by
unit vector in direction
of current I2

unit vector in direction
of current I1
aˆ F12  aˆ2  aˆ1  aˆ R12
unit vector in
direction of force on
I2 due to I1

unit vector in direction
of I2 from I1
Lecture 7
10

The force acting on a current element
I2 dl2 by a current element I1 dl1 is given
by
0 I 2 d l 2  I1d l 1  aˆ R
F 12 
2
4
R12
12

Permeability of free space
0 = 4  10-7 F/m
Lecture 7
11

The total force acting on a circuit C2
having a current I2 by a circuit C1
having current I1 is given by
d l 2  d l 1  aˆ R
 0 I1 I 2
F 12 
2


4 C C
R12
12
2

1
Lecture 7
12

The force on C1 due to C2 is equal in
magnitude but opposite in direction to the
force on C2 due to C1.
F 21   F 12
Lecture 7
13



Ampere’s force law describes an “action
at a distance” analogous to Coulomb’s
law.
In Coulomb’s law, it was useful to
introduce the concept of an electric field to
describe the interaction between the
charges.
In Ampere’s law, we can define an
appropriate field that may be regarded
as the means by which currents exert
force on each other.
Lecture 7
14

The magnetic flux density can be
introduced by writing
0
F 12   I 2 d l 2 
4
C
2

C1
I d l
1
1
 aˆ R12

2
12
R
  I 2 d l 2  B12
C2
Lecture 7
15

where
0
B12 
4

C1
I1d l 1  aˆ R12
2
12
R
the magnetic flux density at the location of
dl2 due to the current I1 in C1
Lecture 7
16

Suppose that an infinitesimal current
element Idl is immersed in a region of
magnetic flux density B. The current
element experiences a force dF given
by
d F  Id l  B
Lecture 7
17

The total force exerted on a circuit C
carrying current I that is immersed in
a magnetic flux density B is given by
F  I  dl  B
C
Lecture 7
18

A moving point charge placed in a
magnetic field experiences a force
given by
F m  Qv  B
Q
v
Id l  Qv
B
The force experienced
by the point charge is
in the direction into the
paper.
Lecture 7
19

If a point charge is moving in a region
where both electric and magnetic fields
exist, then it experiences a total force
given by
F  F e  F m  qE  v  B

The Lorentz force equation is useful for
determining the equation of motion for
electrons in electromagnetic deflection
systems such as CRTs.
Lecture 7
20


The Biot-Savart law gives us the B-field arising
at a specified point P from a given current
distribution.
It is a fundamental law of magnetostatics.
Lecture 7
21

The contribution to the B-field at a
point P from a differential current
element Idl’ is given by
0 I d l   R
d B(r ) 
3
4 R
Lecture 7
22
P
R
Id l 
r
r
Lecture 7
23

The total magnetic flux at the point P
due to the entire circuit C is given by
0 I d l   R
B(r )  
3
4 R
C
Lecture 7
24



Line current density (current) - occurs for
infinitesimally thin filamentary bodies
(i.e., wires of negligible diameter).
Surface current density (current per unit
width) - occurs when body is perfectly
conducting.
Volume current density (current per unit
cross sectional area) - most general.
Lecture 7
25

For a surface distribution of current, the
B-S law becomes
0 J s r  R
B(r )  
ds
3
4 R
S

For a volume distribution of current, the
B-S law becomes
0 J r  R
B(r )  
dv
3
4 R
V
Lecture 7
26

Ampere’s Circuital Law in integral form states
that “the circulation of the magnetic flux
density in free space is proportional to the
total current through the surface bounding
the path over which the circulation is
computed.”
B

d
l


I
0
encl

C
Lecture 7
27
dl
dS
S
I encl   J  d s
S
By convention, dS is
taken to be in the
direction defined by the
right-hand rule applied
to dl.
Since volume current
density is the most
general, we can write
Iencl in this way.
Lecture 7
28
Just as Gauss’s law follows from
Coulomb’s law, so Ampere’s circuital
law follows from Ampere’s force law.
 Just as Gauss’s law can be used to
derive the electrostatic field from
symmetric charge distributions, so
Ampere’s law can be used to derive
the magnetostatic field from
symmetric current distributions.

Lecture 7
29

Ampere’s law in integral form is an
integral equation for the unknown magnetic
flux density resulting from a given
current distribution.
B

d
l


I
0
encl

C
known
unknown
Lecture 7
30


In general, solutions to integral equations must
be obtained using numerical techniques.
However, for certain symmetric current
distributions closed form solutions to
Ampere’s law can be obtained.
Lecture 7
31


Closed form solution to Ampere’s law relies
on our ability to construct a suitable family of
Amperian paths.
An Amperian path is a closed contour to which
the magnetic flux density is tangential and
over which equal to a constant value.
Lecture 7
32
Consider an infinite line current along the zaxis carrying current in the +z-direction:
I
Lecture 7
33
(1) Assume from symmetry and the
right-hand rule the form of the field
B  aˆ B  
(2) Construct a family of Amperian
paths circles of radius  where
Lecture 7
34
(3) Evaluate the total current passing through
the surface bounded by the Amperian path
I encl   J  d s
S
Lecture 7
35
y
Amperian path

x
I
I encl  I
Lecture 7
36
(4) For each Amperian path, evaluate
the integral
 B  d l  Bl
C
magnitude of B
on Amperian
path.
length
of Amperian
path.
 B  d l  B   2 
C
Lecture 7
37
(5) Solve for B on each Amperian path
B
0 I encl
l
0 I
B  aˆ
2 
Lecture 7
38
 B  dl    B d s
C
S
 0 I encl  0  J  d s
S
 Because the above must hold for any
surface S, we must have
  B  0 J
Differential form
of Ampere’s Law
Lecture 7
39

Ampere’s law in differential form implies that
the B-field is conservative outside of regions
where current is flowing.
Lecture 7
40

Ampere’s law in differential form
  B  0 J

No isolated magnetic charges
 B  0
B is solenoidal
Lecture 7
41

Vector identity: “the divergence of the
curl of any vector field is identically
zero.”
    A  0

Corollary: “If the divergence of a
vector field is identically zero, then
that vector field can be written as the
curl of some vector potential field.”
Lecture 7
42

Since the magnetic flux density is solenoidal, it
can be written as the curl of a vector field
called the vector magnetic potential.
 B  0
 B   A
Lecture 7
43

Furthermore, note that the del operator
operates only on the unprimed
coordinates so that
J r  R
1

  J r   
3
R
R
1
    J r 
R
 J r  
 

 R 
Lecture 7
44

Hence, we have
0 J r 

Br    
d
v

4 V  R
Ar 
Lecture 7
45

For a surface distribution of current, the
vector magnetic potential is given by
0
A(r ) 
4

J s r 

d
s
S  R
For a line current, the vector magnetic
potential is given by
0 I d l 
A(r ) 

4 L R
Lecture 7
46


In some cases, it is easier to evaluate the
vector magnetic potential and then use
B
=  A, rather than to use the B-S law to
directly find B.
In some ways, the vector magnetic potential A
is analogous to the scalar electric potential V.
Lecture 7
47


In classical physics, the vector magnetic
potential is viewed as an auxiliary function
with no physical meaning.
However, there are phenomena in quantum
mechanics that suggest that the vector
magnetic potential is a real (i.e., measurable)
field.
Lecture 7
48