Download Magnetostatics

5 – Magnetostatics
Phys4101-E&M
5 – Magnetostatics
Alotted time: 6 lectures
Introduction to the forces and field of the static magnetic field, currents and electric field,
vector potential, description of dipoles.
5.1 Lorentz Force and Currents
5.2 Biot-Savart Law
5.3 Divergence and Curl of B
5.4 Magnetic Vector Potential
5.5 Magnetostatic boundary conditions
5.6 Magnetic dipole
Appendix
Current
density
J  r
A
B
0 I  r  
dl 
4  r
0 I  rˆ
dl 
4 V r 2
 B  0 J
2 A  0 J
B   A
Vector
potential
Ar 
Magnetic
field B  r 
5.1 Lorentz Force and Currents
5.1.1 Magnetic Force and motion of charge

F  q vB
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
Ns
B  1
 1Tesla  1T
 
Cm
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WB   F  dl  0
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1. A particle with charge q and mass m enters a magnetic field B  Byˆ with an initial
velocity v  vxˆ at the origin. Discuss the motion qualitatively. Find the force on the
particle and calculate its trajectory r  t  .
2. A particle with charge q and mass m enters a magnetic field B  Byˆ with an initial
velocity v  vx 0 xˆ  v y 0 yˆ at the origin. Discuss the motion qualitatively. Find the force
on the particle and describe its trajectory.
3. Homework: A particle with charge q and mass m enters a region of space with a
magnetic field B  Byˆ and and electric field E  Ezˆ . The entrance point is the origin,
and the initial velocity is v  vx 0 xˆ  v y 0 yˆ . Discuss the motion qualitatively. Then solve
for the trajectory.
5.1.2 Current
I  v with   dq
Linear current (line charge)
dl
F   dI  B   I dl  B

Magnetic force on wire:


L
K v 
Area current (surface current density)
dI
dl


S
J  v 




F   v  B  da   K  B da
Magnetic force on sruface:
Volume current (volume current density)

L
dI
da
S


F   v  B  d   J  B d
Magnetic force on surface:
S
S
dQ
 J  da   K  dl
dt S
L

d
S J  da  V dt d or   J   t
I
Total current
Continuity equation:
4. A current is traveling through a circular wire of radius R. A magnetic field fills the
region with one half of the circle (see sketch). Calculate the force on the wire in each
case, a, b and c.
B
a)
I
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b)
I
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B
c)
I
B
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5. A thin spherical shell with radius R is rotating at angular speed  around its axis. The
shell carries a homogeneous surface charge σ. Determine the surface charge density
K.
6. (5.5) A current I flows down a wire of radius a.
a) If it is uniformly distributed over the surface, what is the surface current
density K ?
b) If it is distributed in such a way that the volume current density is inversely
proportional to the distance from the axis, what is J ?
5.2 Biot-Savart Law
0 I  rˆ
 I dl   rˆ
dl   0 
2

4 V  r
4 V  r 2
N
Permeability of free space  0  4 107 2
A
ˆ

K r
Surface current
B  0  2 da
4 V  r

J  rˆ
Volume current
B  0  2 d 
4 V  r
Linear current
B
7. Determine the magnetic field in a distance x from a long straight wire carrying a
steady current I.
8. Find the force between two long straight parallel wires, carrying currents I1 and I2,
caused by their magnetic field. The distance between the wires is d.
9. Find the magnetic field for a point on the axis of a circular loop carrying a current I.
Let the radius be a, and the point’s distance from the center of the loop be z.
10. (5.12) Suppose you have two infinite straight line charges λ, a distance d apart,
moving along at a constant speed v. How great would v have to be in order for the
magnetic attraction to balance the electric repulsion? Work out the actual number… Is
this a reasonable sort of speed?
5.3 Divergence and Curl of B
Ampere’s Law
Sources and Sinks
 B  dl   I
 B  da  0
 B  0 J
0 encl
 B  0
11. Use Ampere’s law to find the magnetic field of a straight wire with current I.
12. What is the magnetic field due to a planar current sheet of charge density σ, moving in
x-direction at speed v?
13. What is the magnetic field due to a long, densely wound cylindrical solenoid with
circular cross section?
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5.4 Magnetic Vector Potential
Vector potential
Gauge condition:
B   A
 A  0
0
Boundary condition: A 
r 

14. Find the vector potential for a magnetic field B  B0 zˆ and visualize its properties.
Other accesses to vector potential:
2 A  0 J
Solutions:
A
0 I  r  
0 J  r  
0 K  r 


dl
A

da
A

d 
or
or
4  r
4  r
4  r
15. Determine the vector potential as a function of position for the line indicated in the
square loop.
16. Determine the vector potential for a long solenoid as a function of distance from the
axis of the solenoid.
17. (5.25) By whatever means you can think of, find the vector potential a distance s from
a straight infinite wire carrying a current I. Check that  A  0 and  A  B .
Find the magnetic field inside the wire, if it has a radius R and the current is uniformly
distributed.
5.5 Magnetostatic boundary conditions
Magnetic field lines when passing through a surface current:
parallel component skips
B II  B I  0 K
BI  BII
Comparison:
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normal component is continuous

EII  EI  nˆ
BII  BI  0 K  nˆ
0

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18. For the following cases of an electric or magnetic field encountering a boundary,
sketch the field on the opposite side.
Electric field E
Magnetic field B
Normal
σ(+)
K
incidence
EI
Parallel field
lines
BI
σ(+)
K
EI
BI
Angled incidence
σ(+)
K
BI
EI
Inverted charge
or current
K
σ(-)
BI
EI
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5.6 Magnetic dipole
Magnetic dipole moment:
m  I  da
0
 m  rˆ 
4r 2

B  0 3  3  m  rˆ  rˆ  m 
Dipole magnetic field
4r
19. Visualize the vector potential and the magnetic field of a magnetic dipole using a
circular ring current.
Dipole vector potential
A
20. Find the magnetic dipole moment of a rotating thin shell with angular velocity  and a
surface charge density σ. Show that the magnetic field of the shell out side is exactly
equal to the field of a magnetic dipole.
Appendix
Sorting out the boundary conditions for the magnetic field:
Region I
Region II
The boundary is characterized by a current sheet, which will modify any magnetic fields
present in this space by its own contribution.
1. One can treat this problem without resorting to the split into parallel and normal
components as follows:
The magnetic field will be modified by the magnetic field of the surface current, such that one
can write for the two regions:
BI  Bother  BK , I
Equation 1
BII  Bother  BK , II
Equation 2
The magnetic field of a planar current sheet had been calculated in example 5-12, and can, for
our problem, be written as
1
BK   0 K  nˆ
Equation 3
2

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Here, the normal vector would point away from the surface into the space in which BK is
measured. For the two regions, define nˆII  nˆI  nˆ , with n̂ pointing into region II. Then
equations 1 and 2 become
1
BI  Bother   0 K  nˆ
2
1
BII  Bother   0 K  nˆ .
2
This leads to the difference in magnetic field between regions I and II of

BII  BI  0 K  nˆ





Equation 4
2. One can pursue the split into normal and parallel components and arrive at the same
conclusion as follows:
The field in both regions can be written as a sum of the magnetic field of the current sheet
plus any other magnetic fields, which gives


BI  Bother
 BK,I  Bother


BII  Bother
 BK,II  Bother
for the normal components of the magnetic field. The intrinsic magnetic field of the current
sheet has no normal components. Using  B  da  0 , and a surface of a pill box, one can
conclude that
BII  BI .
For the parallel components
BI  Bother  BK , I
BII  Bother  BK , II
One can use Ampere’s Law
 B  dl
Then
BII
  0 I to move around an Amperian loop.
 B  dl
  BI L  BII L   0 KL .
Since any component in direction of the current will also not
change (the only change occurs in direction of the intrinsic
field of the surface current, one can summarize the findings in
Equation 4, as above.
BI
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