Ch5. ???(Magnetostatics)

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Ch5. ???(Magnetostatics)
5.1 ??????(The Lorentz Force Law) 5.2 ??????(The
Biot-Savart Law) 5.3 ????????(The Divergence and
Curl of B) 5.4 ?????(Magnetic Vector Potential)
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5.1.1 The Lorentz Force Law-??(Magnetic Fields)
Bar magnet
Conducting wire
Parallel wires carrying currents in the same
direction attract each other
Parallel wires carrying currents in opposite
direction repel each other
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5.1.2.1 ??(Magnetic Forces)
The Lorentz force
Additional speed parallel to B
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5.1.2.2 ??(Magnetic Forces)
z
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5.1.2.3 ??(Magnetic Forces)
But the particle started from rest (
), at the origin (
)
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5.1.2.4 ??(Magnetic Forces)
This is the formula for a circle, of radius R,
whose center (0, R?t, R) travels in the
y-direction at a constant speed
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5.1.3.1 ??(Currents)
Example 5.3 A rectangular loop of wire,
supporting a mass m, hangs vertically with one
end in a uniform magnetic field B, which points
into the page in the shaded region. For what
current I, in the loop, would the magnetic force
upward exactly balance the gravitational force
downward?
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5.1.3.2 ??(Currents)
What happens if we now increase the current?
The loop rises a height of h
quB
Rise the loop
qwB
v
u
w
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5.1.3.3 ??(Currents)
Example 5.4
a) A current I is uniformly distributed over a
wire of circular cross section, with radius a.
Find the volume current density J.
b) Suppose the current density in the wire is
proportional to the distance from the axis, J
ks. Find the total current in the wire.
The current in the shaded path
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5.1.3.4 ??(Currents)
The current crossing a surface S can be written as
The total charge per unit time leaving a volume V
is
Because charge is conserved
Continuity equation
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5.2.1 ????(Steady Currents)
Continuity equation
Actually, it is not necessary that the charges be
stationary, but only that the charge density at
each point be constant ( be independent with
time).
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5.2.2 ???????(The Magnetic Field of a Steady
Current)
Example 5.5
Find the magnetic field a distance s from a long
straight wire carrying a steady current I
s
?
?
I
l
dl
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5.2.3 ???????(The Magnetic Field of a Steady
Current)
?
s
?1
?
?2
I
dl
l
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5.2.4 ???????(The Magnetic Field of a Steady
Current)
The field at 2 due to 1 is
into the page
The force on 2 due to 1 is
The force per unit length
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5.2.5 ???????(The Magnetic Field of a Steady
Current)
Example 5.6
Magnetic Field on the Axis of a Circular Current
Loop
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5.2.6 ???????(The Magnetic Field of a Steady
Current)
For surface and volume currents, the Biot-Savart
Law becomes
Problem 5.8
a) Find the magnetic field at the center of a
square loop, which carries a steady current I.
Let R be the distance from center to side.
R
for
Four sides
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5.2.7 ???????(The Magnetic Field of a Steady
Current)
b) Find the field at the center of a rectangular
n-sided polygon, carring a steady current I. Let
R be the distance from center to any side.
for
n sides
c) Check that your formula reduces to the field
at the center of a circular loop, in the limit n
? 8.
n ? 8
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5.2.8 ???????(The Magnetic Field of a Steady
Current)
Problem 5.9
Find the magnetic field at point P for each of
the steady current configurations.
The vertical and horizontal lines produce no
field at P.
b
I
The two quarter-circles
a
P
The two half-lines are the same as one infinite
line
R
I
P
Total
I
The half-circle
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5.3.1.1 ???????? ????(Straight-Line Currents)
The magnetic field of an infinite straight wire
Notice that the answer is independent of s
thats because B decreases at the same rate as
the circumference increases. In fact, it doesnt
have to be a circle any old loop that encloses
the wire would give the same answer. For if we
use cylindrical coordinates (s,?,z), with this
current flowing along the z axis,
This assumes the loop encircles the wire exactly
once.
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5.3.1.2 ???????? ????(Straight-Line Currents)
Suppose we have a bundle of straight wires. Each
wire that posses through our loop contributes
?0I, and those outside contribute nothing.
I4
I3
I enc total current enclosed by the integration
path
I2
I1
If the flow of charge is represented by a volume
current density J, the enclosed current is
Applying Stokes theorem
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5.3.2.1 ????????(The Divergence and Curl of B)
(x,y,z)
The Biot-Savart law for the general case of a
volume current reads
d?
(x,y,z)
Because J doesnt depend on the unprimed
variables (x,y,z)
The divergence of the magnetic field is zero!
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5.3.2.2 ????????(The Divergence and Curl of B)
0
0
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5.3.2.3 ????????(The Divergence and Curl of B)
The x component
For steady currents the divergence of J is zero
This contribution to the integral can be written
On the boundary J 0
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5.3.2.4 ????????(The Divergence and Curl of B)
Ampères law in differential form
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5.3.3.1 ???????(Applications of Ampères Law)
Ampères law in differential form
Ampères law in integral form
Example 5.8
Find the magnetic field of an infinite uniform
surface current , flowing over
the xy plane
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5.3.3.2 ???????(Applications of Ampères Law)
What is the direction of B?
From the Biot-Savart law
Could it have a z-component ? ? no (symmetry)
The magnetic field points to the left above the
plane and to the right below it
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5.3.3.3 ???????(Applications of Ampères Law)
Example 5.9
Find the magnetic field of a very long solenoid,
consisting of n closely wound turns per unit
length on a cylinder of radius R and carrying a
steady current I.
Where N is the number of turns in the length
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5.3.3.4 ???????(Applications of Ampères Law)
Example 5.10
Find the magnetic field of a toroidal coil,
consisting of a circular ring around which a long
wire is wrapped.
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5.3.3.5 ???????(Applications of Ampères Law)
Problem 5.13
A steady current I flows down a long cylindrical
wire of radius a. Find the magnetic field, both
inside and outside the wire, if (a). The current
is uniformly distributed over the outside surface
of the wire. (b). The current is distributed in
such a way that J is proportional to s, the
distance from the axis.
I
a
(a)
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5.3.3.5 ???????(Applications of Ampères Law)
(b)
For s lt a
For s gt a
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5.3.4. ??????????(Comparison of Magnetostatics
and Electrostatics)
The divergence and curl of the electrostatic
field are
Gausss law
The divergence and curl of the magnetostatic
field are
Ampères law
The electric force is stronger than the magnetic
force. Only when both the source charge and the
test charge are moving at velocities comparable
to the speed of light, the magnetic force
approaches the electric force.
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5.4.1.1. ?????(Magnetic Vector Potential)
V electric scalar potential
You can add to V any function whose gradient is
zero
magnetic vector potential
You can add to any function whose curl is
zero
We will prove that
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5.4.1.2. ?????(Magnetic Vector Potential)
Suppose that our original vector potential
is not divergenceless
Because we can add to any function whose
curl is zero
If a function ? can be found that satisfies
Mathematically identical to Poissons equation
In particular, if ? goes to zero at infinity,
then the solution is
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5.4.1.3. ?????(Magnetic Vector Potential)
By the same token, if goes to zero at
infinity, then
It is always possible to make the vector
potential divergenceless
so
This again is a Poissons equation
Assuming goes to zero at infinity, then
For line and surface currents
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5.4.1.4. ?????(Magnetic Vector Potential)
Example 5.11
A spherical shell, of radius R, carrying a
uniform surface charge ?, is set spinning at
angular velocity ?. Find the vector potential it
produces at point .
The integration is easier if we let lie on
the z axis, so that ? is titled at an angle ?.
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5.4.1.5. ?????(Magnetic Vector Potential)
where
Because
We just consider
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5.4.1.6. ?????(Magnetic Vector Potential)
Letting , the integral becomes
If the point lies inside the sphere, Then R
gt r. ?
If the point lies outside the sphere, Then R
lt r. ?
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5.4.1.7. ?????(Magnetic Vector Potential)
If the point lies inside the sphere, Then R
gt r. ?
If the point lies outside the sphere, Then R
lt r. ?
Noting that
For the point inside the sphere
For the point outside the sphere
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5.4.1.8. ?????(Magnetic Vector Potential)
We revert to the original coordinates, in which
coincides with the z axis and the point
is at (r,?,?)
For the point inside the sphere
The magnetic field inside this spherical shell is
uniform!
For the point outside the sphere
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5.4.1.9. ?????(Magnetic Vector Potential)
Example 5.12
Find the vector potential of an infinite solenoid
with n turns per unit length, radius R, and
current I
We cannot use
because the current itself extends to infinity.
Notice that
Since the magnetic field is uniform inside the
solenoid
For s lt R
For an amperian outside the solenoid
For s gt R
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5.4.2.1. ????????(Summary Magnetostatic
Boundary Conditions)
?
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5.4.2.2. ????????(Summary Magnetostatic
Boundary Conditions)
For an amperian loop running perpendicular to the
current
Perpendicular to the current
For an amperian loop running parallel to the
current
Parallel to the current
Where is a unit vector perpendicular to the
surface, pointing upward.
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5.4.2.3. ????????(Summary Magnetostatic
Boundary Conditions)
The vector potential is continuous across any
boundary
For an amperian loop of vanishing thickness
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5.4.3.1. ?????????(Multipole Expansion of the
Vector Potential)
A multipole expansion which is an approximate
formula and valid at distant points) for the
vector potential of a localized current
distribution.
I
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5.4.3.2. ?????????(Multipole Expansion of the
Vector Potential)
monopole
dipole
quadrupole
The magnetic monopole tern is always zero
Let , where is a constant
vector.
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5.4.3.3. ?????????(Multipole Expansion of the
Vector Potential)
Let
where
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5.4.3.4. ?????????(Multipole Expansion of the
Vector Potential)
Example 5.13
Find the magnetic dipole moment of the
Bookend-shaped loop shown in Figure below. All
sides have length w, and it carries a current I.
w
w
w
The wire could be considered the superposition of
two plane square loops shown in Fig. 5.53 of text
book.
The combined (net) magnetic dipole moment is
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5.4.3.5. ?????????(Multipole Expansion of the
Vector Potential)
The magnetic dipole moment is independent of the
choice of origin.
The magnetic field of a (pure) dipole is easier
to calculate if we put the dipole moment at the
origin and let it point in the z-direction.
This is identical in structure to the field of an
electric dipole!