Download Chapter 9 THE MAGNETIC FIELD

Chapter 9 THE MAGNETIC FIELD
• Introduction
• Magnetic field due to a moving point charge
• Units
• Biot-Savart Law
• Gauss’s Law for magnetism
Figure 1 The magnetic field due to a point charge q
moving with velocity v.
• Ampère’s Law
• Maxwell’s equations for statics
• Summary
INTRODUCTION
Last lecture introduced the concept of a magnetic field
and the magnetic dipole. It was seen that the magnetic
dipole has a north-seeking and a south-seeking pole.
Opposite poles attract and like poles repel. Magnetic
poles cannot be isolated in contrast to the charges of
an electric dipole.
Oersted’s discovery that an electric current produces a magnetic field circling clockwise around the
electric current is especially important in that it shows
that electricity and magnetism are directly related.
The Lorentz force describes the force acting on a
charge in both electric and magnetic fields.
−
→
→ → −
−
→
F = ( E + −
v × B)
The Lorentz force was used to derive the force acting
on an electric current in a magnetic field. The unusual
feature of the magnetic force acting on moving charges
→
−
→
is that the force is perpendicular to both B and −
v
Summing the magnetic forces for the  moving
charges in an infinitessimal volume  and using the
→
−
→
fact that j =  −
v leads to the magnetic force is
→ −
−
−
→
→
F = j × B
For a conductor of cross sectional area  and length
→
−
→
−
 the volume  = . Then since I =  j , then
the magnetic force on a circuit carrying a current  in
a magnetic field  is
I
I
→ −
−
→
−
→ −
−
→
→
F =
I × B = 
dl × B


The general features of the magnetic field produced
by electric currents has been discussed qualitatively.
The next stage is to derive the magnitude as well as
the direction of the magnetic field produced by moving
charges and currents. The discussion of the electrostatic force started with the experimental facts, condensed into Coulomb’s Law plus superposition, and
from these derived Maxwell’s Equations for electrostatics. The concepts of flux and circulation are required
to express fully the laws of electromagnetism. For electrostatics the concept of flux led to Gauss’s law while
the concept of circulation led to the proof that the electric field is conservative allowing use of the concept of
electric potential. For magnetism it is logical also to
start with the experimental facts for the magnetic force
and from these derive the flux and circulation of the
→
−
vector B field. Then it is possible to make a comparison of the difference between the laws for electrostatics
and magnetism.
There are two approaches to introducing the basic experimental facts of the magnetic force. One approach is to use Coulomb’s Law plus Einstein’s Theory
of Relativity to derive directly the magnetic field produced by a moving point charge. A second approach
is to start with the Biot Savart Law to define the magnetic field due to an infinitessimal element of current.
These will be introduced and then will be used to derive Gauss’s law describing the flux of the magnetic
field, while the concept of circulation will lead to Ampère’s Law.
MAGNETIC FIELD OF A MOVING
POINT CHARGE
Coulomb’s Law provides the basis for calculating the
electric field. An equivalent law is needed for calculation of magnetic fields. The most fundamental relation is the magnetic field produced by a moving point
→
charge. For a point charge moving with a velocity −
v
→
−
the induced magnetic field B is:
→
−
→
 −
v ×b
r
B= 0
2
4 
67
Figure 2 Magnetic field due to an electric current in a
circuit.
Figure 3 Geometry for the magnetic field due to an
infinitely long straight current.
→
−
→
Note that B is proportional to  and −
v as well as
varying inversely with the square of the distance .
→
−
→
→
The direction of B is perpendicular to both −
v and −
r
→
−
→
−
and is a maximum when r is perpendicular to v  The
→
magnetic field circles clockwise around the vector −
v
This formula can be derived from experimental data.
However, also it can be derived from Coulomb’s law
using Einstein’s Theory of Relativity.
charges in a wire assuming that the Principle of Superposition applies. For  charges per unit volume, the
→
−
net magnetic field ∆ B due to an element of volume
 is  times the field due to each charge. That is:
UNITS
The constant  has been chosen such that
0 ≡ 4 × 10−7  = 4 × 10−7 2
exactly and is called the permeability of free space.
0
That is, 4
is chosen to be exactly 10−7 in the SI
system of units as will be discussed later. The factor 4 was inserted to simplify Ampère’s law as will
be discussed later. Since, in the MKS system, the
Lorentz force is defined in Newtons and velocity in
meters/second, defining the constant 0 fixes the unit
of charge , the coulomb. However, in Coulomb’s Law
the Coulomb unit of charge was defined in terms of
1
the constant 4
 Clearly, the constants  and  are

related .
The magnetic field due to a moving charge is the
−
→
simplest system for definition of the B field, but in
practice it is more useful to use the expression for the
→
−
B field due to an element of a circuit carrying an electric current , which is called the Biot Savart law.
BIOT SAVART LAW
One month after Oersted’s discovery of the magnetic
field produced by an electric current, Biot and Savart
determined experimentally the magnetic field due to
a long straight wire. This result also can be derived
by summing over the magnetic field due to the moving
68
→
−
→
−−→ 0  −
0 j × b
v ×b
r
r
∆B =
 =

2
2
4

4 
→
−
→
since j =  −
v . The magnetic field at a distance r
from an element of a conductor of cross sectional area
 and length  carrying current , can be calculated
since the volume element is  = , and knowing
→
−
→
−
that I =  j  Thus:
→
−
→  I ×b
−
r
∆B = 0

2
4 
Integrating over a closed circuit gives
−
→

B= 0
4
I


−
→
I ×b
r

2
→
−
→
−
→
−
Another convention writes I  =  l where l carries the information as to the direction of the current
flow. Integrating over a complete closed circuit, using
this notation, gives that the total magnetic field at the
point r from a circuit carrying current , is:
−
→

B= 0
4
I
→
−
 dl × b
r
2



This is called the Biot Savart Law. It is equivalent
to Coulomb’s Law in that one can compute, by integration, the magnetic field due to any shape currentcarrying circuit. Let us consider an application of the
Biot Savart law. Note that the Biot Savart Law has
been written assuming a right-handed coordinate system.
Field due to an infinitely long straight current
Orient the wire along the  axis as shown in figure 3.
From symmetry, the magnetic field at a given distance
 from the wire must look the same anywhere in the
Figure 4 The magnetic field produced by a long straight
electric current.
 −  plane. Thus the magnetic field will be computed
at the point  =  and  = 0. The contribution to the
→
−
 field from the element l is given by:
Since
→ 
−
sin φ b 0
cos θ b
∆ B = 0  2 k
=
 2 k
4

4

 =
then
 =
Figure 5 Magnetic field on the axis of a circular current
loop.
due to the element  or the current loop
→
−
dB =




cos2 
=
→
−
Then the components of d B along the  and  axes
are
and

cos 
Integrating over the complete wire gives:
 =  cos  =
→
−
0
 

4 (2 + 2 ) (2 + 2 )12
 =  sin  =
→
−
 

0
2
2
2
4 ( +  ) ( + 2 )12
=
−
→
B =
=
−
→
B =
0

4
Z

2
−
2
Z

2
 cos2 
b
cos k
cos2  2
0 b
cos 
k

4

−
2
→ b
−
0 I × R
2 
where the cross product is included to carry the information of the direction of the magnetic field. That
is, the magnetic field circulates clockwise around the
current as shown. Note that for a 25 current, the 
field at  = 5 is 10−4  ≈ 1
Field on the axis of a circular current loop
A current loop is called a magnetic dipole. It is of
interest to calculate the magnetic field on the axis of
the magentic dipole. Calculation of the cross product
→ −
−
l×→
r can be written in terms of cartesian coordinates.
As shown in figure 5, assume that the circle is in the
 −  plane.
Since  and  are perpendicular, then the Biot
Savart law can be used to evaluate the magnetic field
→
−
r
0  dl × b
4 2
→
−
0  dl × b
r
4 (2 + 2 )
and
By symmetry for the whole loop only the  component is non-zero, thus
Z
0 

 =
3
4
2
( + 2 ) 2
Z
0 

=

4 (2 + 2 ) 32
But the circumference of a circle equals 2, thus
−
→
 
2
bi
B= 0
2 (2 + 2 ) 32
This gives the  field on the axis of the magnetic dipole
by the brute force approach. Superposition of the 
field can be used to obtain the  field for a magnetic
dipole having  turns to be
−
→
 
2
bi
B= 0
2 (2 + 2 ) 32
Figure 6 illustrates the magnetic field along the axis
of a circular loop of  turns current .
69
Figure 8 Gaussian surface for a long straight current I.
GAUSS’S LAW FOR MAGNETISM
Figure 6 Graph of magnetic field along the axis of a
current loop on  turns. When    then the field
falls off as 13 
It was seen that Gauss’s Law for electrostatics is of
considerable theoretical importance as well as being
powerful for calculating electric fields for symmetric
systems. That is, the net electric flux out of a closed
surface is:
Φ =
I
→
− −
→
1
E · S =

0
 
Figure 7 Helmholtz coils used to cancel the earth’s
magnetic field at the center of the current loops.
Note that at the distance  =  the  field falls
to one half of the maximum value at  = 0. This is
exploited by the Helmholtz arrangement of two such
magnetic dipoles, shown in figure 7, where two identical coils separated by one radius produce a very uniform magnetic field along the axis near the centre of
the coils.
The off-axis field for the magnetic dipole is more
complicated and will be derived later. The Biot Savart
law can be used to calculate, by brute force, the  field
around any current-carrying circuit. Unfotunately, evaluating the integrals for most geometries can be challenging.
70
Z


τ
Remember that this is a statement of the fact that
Coulomb’s law states that the electric field has a r2
dependence. It states that the net flux out of a closed
surface equals the enclosed charge times a constant 10 
This is independent of the shape or size of the closed
surface because the the 12 dependence of Coulomb’s
law.
Gauss’s Law for magnetism is given by computing the net flux of the magnetic field out of a closed
Gaussian surface. Consider the special case of a concentric cylinder surrounding a long straight current
shown in figure 8. As calculated with the Biot Savart
Law, the magnetic field is tangential to the surface of
the concentric cylinder around the current, and also is
tangential to the ends of the cylinder. Thus there is
no net flux out of the cylinder, that is,
#
"Z
−−→
−−→
I
Z
→
→ −
−

I×b
r
I×b
r b
B · S =
·b
r +
· I
2
2  2

 
= 0
since the surface vectors are perpendicular to the cross
product over all of the surface of the cylinder. It can be
shown using the Biot Savart law that this is a general
property of magnetostatics. Thus the most general
form of Gauss’s Law for magnetostatics is:
I
→
→ −
−
Φ =
B · S = 0

 
The net magnetic flux is independent of the size or
shape of the closed surface, as expected since the field
due to a point charge with velocity v has a 12 depen→
−
dence. Moreover the net flux is zero because the B
field is tangential to b
r
Figure 9 Lines of  intersecting a Gaussian surface.
Any flux tube entering the surface must exit if the lines
of  are continuous.
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 as illustrated in figure 9.
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. For that purpose one has to turn to the relation for the circulation
of the magnetic field.
AMPÈRE’S LAW
→
−
In electrostatics it was found 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.
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
Figure 10 Concentric circular line integral around a
long straight current.
Figure 11 Arbitrary shaped closed loop enclosing long
straight conductor.
−
→
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 current is assumed to flow in the direction
given by the right-hand rule relative to the direction
of the closed line integral.
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  as a boundary. Consider an
arbitrary shaped closed loop enclosing a long straight
conductor as shown in figure 11. Note that for an
element of line  at a radius  from the line current:
→
→ −
−
B · dl =  cos  = 
→
−
since B is tangential. Therefore for a long straight
conductor one obtains:
I
I
→  
→ −
−

B · dl = 
2


I
I
→  
→ −
−
B · dl = 

2

71
Figure 12 Closed loop 1 enclosing the current carrying
conductor and the closed loop 2 not enclosing the
current carrying conductor.
charge inside the volume bounded by 1 + 2 , then
the net current flowing into the volume must equal
R−
→ −
→
the net outflow of current. That is,
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 .
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
STATICS
It is interesting to compare and contrast the flux and
circulation equations we have derived for static electric
and magnetic fields.
Figure 13 Closed loop and surface bounded by closed
loop as used by Ampère’s Law.
Flux
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 

= 0 (Enclosed current)
If the conductor is outside the closed loop in figure 12,
e.g. 2  then the angle integral equals zero.Again it is
assumed that the current flows in a direction given by
the right-hand rule for the line integral.
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:
I
Z
→
→ −
−
→
→ −
−
 B · dl = 0  j · S




where the surface is bounded by the closed loop C. It is
−
→
important that the direction of the line integral and S
be given by the right-hand rule. 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 .
As shown in figure 13, take surfaces 1 and 2 both
bounded by . If there are no sources or sinks of
72
Circ.
Electrostatics
H→
R
→
− −
E S = 1 
H−
→
→−
E  l = 0
Magnetostatics
H−
→
→−
BS Z= 0
H−
→
−
→ −
→
→−
B l = 0 j · S
These are the Maxwell equations for statics. The
flux relations show that the electrostatic field has nonzero flux for a Gaussian surface enclosing charge because charges are sources and sinks of the electric field,
whereas the magnetic field has zero flux out of a Gaussian
surface because there are no sources or sinks of magnetic field.
The circulation relations show that the static electric field has zero circulation, because the electric field
for a point charge is radial, whereas the circulation of
the magnetostatic field has a non-zero circulation if it
encloses an electric current. Thus, in contrast to the
static electric field which is circulation free, the magnetostatic field is flux free.
For statics the electric field and magnetic field are
unrelated by the Maxwell equations. It will be shown
later that the circulation equations lead to coupling
of the magnetic and electric fields for time-dependent
systems.
SUMMARY
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



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 
Gauss’s law for magnetism gives that the total magnetic flux out of a closed surface is:
I
→
→ −
−
B · S = 0
Φ =

 
The circulation of the magnetic field leads to Ampère’s
law.:
I
Z
→ −
−
→
→
→ −
−
B
·
dl
=

0  j · S





This is especially useful for calculating magnetic fields
for systems possessing symmetry.
It is interesting to compare and contrast the flux
and circulation equations we have derived for static
electric and magnetic fields.
Flux
Circ
Electrostatics
R
H→
→
− −
E S = 1  
H →
→
− −
E  l = 0

Magnetostatics
H−
→
→−
BS = 0
H →
→
− −
B l = 0

Z



− −
→
→
j · S
These are the Maxwell equations for statics. The
flux relations show that the electrostatic field has nonzero flux for a Gaussian surface enclosing charge because charges are sources and sinks of the electric field,
whereas the magnetic field has zero flux out of a Gaussian
surface because there are no sources or sinks of magnetic field.
The circulation relations show that the static electric field has zero circulation, because the electric field
for a point charge is radial, whereas the circulation of
the magnetostatic field has a non-zero circulation if it
encloses an electric current. Thus in contrast to the
static electric field which is circulation free, the magnetostatic field is flux free.
For statics the electric field and magnetic field are
unrelated by the Maxwell equations. It will be shown
later that the circulation equations lead to coupling
of the magnetic and electric fields for time-dependent
systems.
The next lecture will discuss further applications of
Ampère’s law, magnetic forces and then review what
we have done so far this term.
Reading assignment: Giancoli, Chapter 28.1—
28.6.
73