Chapter 10 AMPÈRE’S LAW
Gauss’s law for magnetism gives that the total magnetic flux out of a closed surface is:
• Introduction
• Ampère’s Law
• Maxwell’s equations for static systems
Φ =
• Applications of Ampère’s Law
• Forces between circuits
• Summary
INTRODUCTION
The last lecture discussed magnetic fields produced by
moving charges and electric current. The magnetic
field due to a charge moving at velocity v at a distance
r from the charge was stated to be:
→
−
→
 −
v ×b
r
B= 0
4 2
This magnetic field circles around the velocity vector of
the moving charge in a right-handed coordinate frame.
In the SI system of units, the distances are in meters,
force in Newtons while the constant has been chosen
0
to be 4
≡ 10−7 
Summing the magnetic field from all the moving
charges in a closed circuit leads to the Biot Savart
Law which gives the field at a point r from a circuit
carrying current I as:
−
→

B= 0
4
I
→
−
 dl × b
r
2



Again the magnetic field is perpendicular to r and the
element of the circuit dl The Biot Savart Law provides
a brute-force way of computing the magnetic field due
to a current-carrying circuit. This was used to derive
the magnetic field around a long straight current. The
computed magnetic field circles clockwise around the
current. That is, if the right thumb point along the
current then the right-hand fingers point in the direction of the magnetic field.
Figure 1 The magnetic field due to a point charge q
moving with velocity v.
I

 
→
− −
→
B · S = 0
Gauss’s Law for magnetism is a statement that there
are no magnetic monopoles, that is, there are no sources
→
−
or sinks of magnetic field and therefore the lines of B
→
−
are continuous. Since the lines of B are continuous,
then the number entering a closed surface must equal
the number leaving the surface. Gauss’s law for magnetism is useful for limiting the form of the magnetic
field. For example, there cannot be a radial component
→
−
to B around a current element. However, Gauss’s law
for magnetism is not useful for calculating the strength
→
−
of the B field. Here one has to turn to the relation for
the circulation of the magnetic field.
AMPÈRE’S LAW
→
−
In electrostatics we had that the circulation of E around
a closed loop is zero. That is:
I


→
− −
→
E · l = 0
The statement that circulation of the electric field is
zero reflects the fact that the electric field of a point
charge is radial. It states that the electric field is conservative which allows use of the powerful concept of
electric potential. In magnetism, one can use the Biot
→
−
Savart law to relate the circulation of B around a
closed loop to the current flowing through the loop,
leading to Ampère’s law as shown below.
Figure 2 The magnetic field produced by a long straight
electric current.
75
Figure 3 Concentric circular line integral around a long
straight current.
Figure 5 Line integral 1 that encloses the currentcarrying conductor, and a line integral 2 that does not
enclose the current.
Therefore for a long straight conductor one obtains:
I
I
→  
→ −
−

B · dl = 
2


I
Figure 4 Arbitrary shaped closed loop enclosing long
straight conductor.
Concentric circle around long straight conductor
For simplicity, consider the special case of a magnetic
field around a long straight current . The Biot Savart
law gave that:
→
−
−
→
 I × b
r
B=
2 
The circulation, given by the line integral for a concentric circle  of radius  taken in the direction of
→
−
B, is
I
→  
→ −
−
B · dl =  2 = 0 
2 

Thus, for this special case, we obtain Ampère’s Law:
I
→
→ −
−
B · dl = 0 (Enclosed current)
where the line integral is taken clockwise with respect
to the direction of the current 
Arbitrary closed loop around long straight conductor
The Biot Savart law can be used to prove that this is
true for any current distribution through any surface
having the closed loop C as a boundary. Consider an
arbitrary shaped closed loop enclosing a long straight
conductor as shown in figure 4. Note that for an element of line :
→
→ −
−
B · dl =  cos  = 
76
→  
− −
→
B · dl = 
2

I

H
since the  factor cancels. Note that the integral  =
2 if the closed loop encloses the origin, e.g. 1 , that
is
I
→
→ −
−
B · dl = 0 

If the conductor is outside the closed loop, e.g. 2 
then the angle integral equals zero.
The more general form of Ampère’s Law is written
in terms of the current density using the fact that
Z
→ −
−
→
=
j · S
 
leading to the relation:
Z
I
→
→ −
−
→
→ −
−
B
·
dl
=

0  j · S





where the surface is bounded by the closed loop . It
is important that the direction of the line integral and
−
→
S be given by the right-hand rule, that is, the line
integral is taken in a clockwise direction with respect
−
→
to the direction of S. Note that this proof implicitly
assumes that the magnetic fields produced by different
currents superpose, that is the Principle of Superposition has been assumed.
There is an infinite number of surfaces that can
be drawn that are bounded by one closed loop C.
As shown in figure 6 take surfaces 1 and 2 both
bounded by . If there are no sources or sinks of
charge inside the volume bounded by 1 + 2 , then,
by charge conservation, the net current flowing into the
volume must equal the net outflow of current. That is,
R−
→ −
→
j · S is the same for both surfaces. Thus the net
current flowing though the closed loop  is independent of the shape of the surface bounded by the closed
loop .
Figure 6 Closed loop and surface bounded by closed
loop as used by Ampère’s Law.
Ampère’s law is of considerable theoretical importance beyond that of the Biot Savart law from which it
was derived. Also Ampère’s Law provides an easy way
to compute the magnetic field for systems possessing
symmetry. Unfortunately, there are only a limited set
of cases where is is possible to use symmetry to find a
→
→ −
−
curve for which B · l is constant.
MAXWELL’S EQUATIONS FOR
STATIC SYSTEMS
It is interesting to compare and contrast the flux and
circulation equations, Maxwell equations, we have derived for static electric and magnetic fields.
Figure 7 Graph of B versus r for a wire of radius a
carrying uniform current I.
APPLICATION OF AMPÈRE’S LAW
Ampère’s Law allows calculation of magnetic fields for
systems having a symmetry that defines a closed path
→
→ −
−
for which B · l in constant. Let us consider a simple
example having cylindrical symmetry.
Long straight current I of radius a.
First consider ra:
By symmetry the field must have axial symmetry.
From Gauss’s Law we know that the field must be
tangential since there is no net flux out of a concentric
→
−
Electrostatics
Magnetostatics
cylindrical surface. Thus knowing that the B field
R
H
H−
−
→
→
→
−
→−
BS Z= 0
E S = 1 
Flux
is tangential to concentric circles about the current
H−
H−
→
→
→ −
−
→ simplifies evaluation of Ampère’s law. Take the line
→−
→−
Circ.
E  l = 0
B l = 0 j · S
integral for a concentric closed circle of radius  and
use Ampère’s Law:
I
→
→ −
−
The flux relations show that the electrostatic field
B · dl = 0 (Enclosed current)
has non-zero flux for a Gaussian surface enclosing charge

I
because charges are sources and sinks of the electric
→
→ −
−
B · dl = 2 =  
field, whereas the magnetic field has zero flux out of a

Gaussian surface because there are no sources or sinks
→
−
of magnetic field.
Thus the B field has a magnitude given by:
The circulation relations show that the static elec 
tric field has zero circulation, because the electric field
= 0
2

for a point charge is radial, whereas the circulation of
the magnetostatic field has a non-zero circulation if it This is the same as was computed using the Biot Savart
law outside a long thin straight current carrying conencloses an electric current.
For statics the electric field and magnetic field are ductor.
For ra:
unrelated by the Maxwell equations. It will be shown
The same arguments apply except that the enclosed
later that the circulation equations lead to coupling
current
passing through the concentric circle now only
of the magnetic and electric fields for time-dependent
 2
contains
a current  = 
2  Thus using Ampère’s
systems.
law we have:
This lecture will focus on use of Ampère’s Law to
derive the magnetic field for symmetric systems and
2
2 = 0 2 
magnetic forces between circuits.

77
Figure 8 Cylindrical coaxial conductors.
That is:
0 

2 2
Thus the magnitude of the field has the form shown in
figure 7.
The direction of the B field is given assuming that
the normal to the surface bounded by the circleI  is
→
−
B·
in a right-handed coordinate system. That is, if

→
−
→
−
dl  0 and dl is taken in a clockwise direction, then
the current must flow into the face of the clockwise
circle.
=
Cylindrical concentric coaxial conductors
A frequently-used system is that of cylindrical coaxial
conductors as illustrated in figure 8. Assume that a
current I flows in the inner conductor and returns via
the outer cylindrical conductor. The magnetic field
between the concentric cylinders can be obtained using
the line integral along a concentric circle of radius  
   As shown above, using Ampère’s law
I
→
→ −
−
B · dl = 2 =  

→
−
Thus the B field has a magnitude given by:
=
0 
2 
that there are  turns carrying current . To calcuH−
→
→ −
late  we evaluate the line integral B · l around a
circle of radius r centered at the enter of the toroid.
Consider the three cases:
Inside the solenoid, a  r  b:
I
→
− −
→
B · l = 2 = 0  

2
Note that the B field is clockwise since the current is
assumed to flow down into the inside of the loop.
Outside the solenoid with r  a:
Here the net current through the surface bounded
by the circular line integral path is zero, therefore B=0
Outside the solenoid with r  b:
Here the net current through the surface bounded
by the circle is zero since as much current flows in as
out. Thus again the  field outside of the solenoid is
zero.
 = 0
Infinite straight tightly-wound solenoid
One approach to solve this is to take the limit of the
toroid when the radius  → ∞ To eliminate the infinite number of turns and length, express the field of a
toroid in terms of the number of turns per unit length

 = 2
. That is:
 = 0 
(    )
For    the net current is zero through a closed
concentric circle surrounding the outside of the coaxial
cable, thus the net magnetic field outside must be zero.
The magnetic field for    is given as shown in
figure 7 assuming the current in the inner conductor is
uniformly distributed. If the current density is only on
the surface of the inner conductor than the magnetic
field for    will be zero.
Magnetic field of a tightly-wound toroid.
By symmetry, and knowing that the net magnetic flux
is zero, implies that the magnetic field must be tangential to circles concentric with the doughnut. Assume
78
Figure 9 A toroid with N turns carrying current I.
Note that this is independent of the radius  of the
toriod. In particular, it gives the same answer for a
long straight solenoid for which  → ∞
Another way to solve the infinite solenoid is to integrate the line integral over the closed rectangular loop
shown in figure 10. Assuming that the magnetic field
is zero on the outside of the solenoid, then applying
Ampère’s law for a loop of length  that encloses 
turns gives
I
→
→ −
−
B · dl =  =  

 =  
which is the same as given previously.
is used. It can be shown that this leads to the relation
³→ −→´
I I −
dl · dl
−
→

F = − 0  
rc

2
4

 
Figure 10 Infinite straight solenoid.
Figure 11 Forces between current-carrying circuits.Forces between current-carrying circuits.
FORCES BETWEEN TWO CLOSED
CIRCUITS
Earlier it was shown that the force on a circuit a in a
magnetic field produced by circuit b is given by integrating the force per unit length over the whole circuit

−
→
F = 
I

−→ −→
l × B
The Biot Savart Law can be used to calculate the 
field at circuit a due to circuit b.
I −→
−→ 0  dl × rc

B =
2
4 

Combining these gives the force on circuit  due to
circuit  as:
→
−→
I I −
−
→ 0
dl × dl × rc

F =
 
2
4

 

This magnetic force between circuits could be used as
the definition of the magnetic field, that is, one could
reverse to above steps to factor the magnetic force
by postulating a magnetic field , given by the Biot
Savart Law relation plus the force on a current circuit
in such a magnetic field.
This formula does not appear to be symmetric with
respect to interchange of  and , which is required to
satisfy Newton’s third law of motion.The symmetry is
more apparent if the vector identity relation
→ ³−
−
→ −
→´ −
→ ³−
→ −
→´ ³−
→ −
→´ −
→
A× B×C =B A·C − A·B C
→
−
→
−
which is symmetric with respect to dl and dl . Note
that if both circuits have the current rotating in the
→ −→
−
same direction, that is, dl · dl is positive, then the
circuits attract. Moreover the force on circuit  due to
circuit  is equal and opposite as required by Newton’s
Laws of Motion.
This complicated double integral is of importance
since it defines the Ampere. In the SI system of units,
the distances are in meters, force in Newtons while
0
the constant has been chosen to be 4
≡ 10−7  Having
defined the Ampere, then the Coulomb is defined as
the charge due to a current flow of one ampere for one
second. Having defined the Coulomb, then Coulomb’s
Law fixes the constant 0  That is, the arbitrary choice
of 0 fixes the Ampere, Coulomb and 0 The magnetic force relation was chosen to define the units for
Ampere and Coulomb, rather than using Coulomb’s
force law, because experimentally the forces between
circuits and electric currents can be measured accurately, whereas it is difficult to measure charge .
The integral is dimensionless and always is less
than unity except for the special case where the circuits overlap perfectly. Note that you will be required
to evaluate this double integral only for simple systems
for P114. The general form is shown only to illustrate
how magnetic forces between current-carrying circuits
are computed.
Force between infinite parallel conductors
The special case of the magnetic force between two
infinitely long straight parallel wires is simple to compute. It was shown that the magnetic field due to an
infinitely long straight current I at a distance R from
the wire has a magnitude:
−
→
 
B= 
2 
and points in the tangential direction. Substitution of
this into the magnetic force relation for the force on a
length dl in the parallel wire of circuit  is:
−
→ −
→ −
→
  
∆F = I × B  = 

2 
For example, a household circuit carrying 10 in
parallel wires 3 apart is 610−3  Note that the
force is repulsive for opposing currents whereas it is
attractive for currents in the same direction. This
attractive phenomenon explains the Pinch Effect in
plasma discharges. For very large currents, the plasma
is pinched to a small diameter tube because of the
79
due to saturation of iron at high magnetic fields. However, there are two new problems, the first is generating mega-ampere currents, the second is the enormous forces between the conductors. The force per
unit length  is given by
−
→
→ −
−
→
F = I × B = 106 
Figure 12 Magnetic force between two infinite straight
parallel currents.
that is, the repulsive force is 100 tons-weight per meter!.
The accelerator was never built because of problems developing a source of mega ampere currents.
This project was nicknamed the ”White Oliphant”.
Modern charged particle accelerators based on magnets, such as the cyclotron, use superconductors to
create large magnetic fields with minimal power consumption. The primary use of power for the magnet
is to run the refigerator. This is in contrast to the
several megawatts of power consumed by a conventional cyclotron magnet. These magnets have to be
very strong mechanically to withstand the magnetic
forces acting on the coils.
Figure 13 The magnetic field produced by two counterrotating current-carrying conductors.
SUMMARY
magnetic force on the current due to the magnetic
field produced by this current. The hope of generating
power by plasma fusion depends on the Pinch effect to
generate a high plasma density sufficient to produce a
self-sustaining fusion reaction.
Oliphant’s accelerator
In the 19500 Professor Oliphant wished to construct a
high energy proton accelerator without having to build
a very large conventional iron-cored magnetic ring. He
thought of a new scheme for generating a very large
magnetic field required to confine the protons to a circular orbit while they are accelerated. Rather that
building a conventional iron-cored magnet, he conceived of using two counter-rotating conductors each
carrying about 106  as illustrated in the figure. A
vacuum cavity was to be built between the two conductors around the circle through which the proton
beam would be accelerated in the presence of the high
magnetic field due to the circulating current.
The magnetic field in the overlap region of the circular cross section conductors can be calculated assuming superposition of two equal and opposite current
densities, which cancel in the overlap region. Assuming cross sectional radii of  = 04 and  = 106 ,
then the magnetic field in the overlap region is given
by Ampère’s law to be
≈2
 
= 1
2
This magnetic field is adequate to confine the planned
circulating proton beam, and eliminates the limitation
80
The magnetic field due to a moving charge is:
→
−
→
v ×b
r
 −
B= 0
4 2
The Biot Savart Law gives the field at a point r from
a circuit carrying current I as:
−
→

B= 0
4
I
→
−
 dl × b
r
2


Maxwell equations for static electric and magnetic
fields are
Electrostatics
H→
R
→
− −
E S = 1 
H−
→
→−
E  l = 0
Magnetostatics
H−
→
→−
BS Z= 0
H−
→
→ −
−
→
→−
B l = 0 j · S
The forces between magnetic circuits was derived.
The force on circuit  due to circuit  as:
³→ −→´
I I −
dl · dl
−
→

F = − 0  
rc

2
4

 
This complicated double integral is of importance since
it defines the Ampere. In the SI system of units, the
distances are in meters, force in Newtons while the
0
constant has been chosen to be 4
≡ 10−7 
Reading assignment: Giancoli, Chapter 28.128.6.