1. Electro-Magneto-Statics M. Berrondo
Physics 441
Electro-Magneto-Statics
M. Berrondo
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2. 1. Introduction Electricity and Magnetism as a single field
(even in static case, where they decouple)
Maxwell: * vector fields
* sources (and sinks)
Linear coupled PDE’s
* first order (grad, div, curl)
* inhomogeneous (charge & current distrib.)
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3. interpretation of equations and their solutions
Tools
Physics
Math
trigonometry
vectors (linear combination)
dot, cross, Clifford
vector derivative operators
Dirac delta function
DISCRETE TO CONTINUUM
INTEGRAL THEOREMS:
* Gauss, Stokes, FundThCalc
cylindrical, spherical coord.
LINEARITY
trajectories: r(t)
FIELDS: * scalar, vector
* static, t-dependent
SOURCES: charge, current
superposition of sources =>
superposition of fields
unit point sources
Maxwell equation
field lines
potentials
charge conservation
interpretation of equations and their solutions
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4. 2. Math. Review sum of vectors: A , B => A + B
dilation: c , A => c A
linear combinations: c1 A + c2 B
scalar (dot) product:
A , B => A  B = A B cos ( ), a scalar
where A2 = A  A (magnitude square)
cross product:
A , B => A  B = n A B |sin( ) |, a new vector,
with n  A and B, and nn = n2 = 1
orthonormal basis: {e1 , e2 , e3 } = {i , j , k}
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5. Geometric Interpretation: Dot product
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6. Cross Product and triple dot product
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7. Rotation of a vector (plane)
Assume s in the x-y plane
Vector s‘ = k x s
Operation k x rotates s by 90 degrees
k x followed by k x again equivalent to multiplying by -1 in this case!! y s’ s x
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8. Rotation of a vector (3-d)
n unit vector: n2 = 1 defines rotation axis  = rotation angle vector r  r’ where
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9. Triple dot product
is a scalar and corresponds to the (oriented) volume of the parallelepiped {A, B, C}
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10. Triple cross product The cross product is not associative!
Jacobi identity:
BAC-CAB rule:
is a vector linear combination of B and C
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11. Inverse of a Vector
0 1 | | n-1 = n as a unit vector along any direction in R2 or R3
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12. Clifford Algebra Cl3 (product)
starting from R3 basis {e1 , e2 , e3 }, generate all possible l.i. products 
8 = 23 basis elements of the algebra Cl3
Define the product as:
with
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13. 1 ê1 ê2 ê3 iê1= ê2ê3 iê2= ê3ê1 iê3=ê1ê2 2 i = ê1ê2ê3 3 R i R R3 i R3
scalar ê1 ê2 ê3
vector
iê1= ê2ê3
iê2= ê3ê1
iê3=ê1ê2 2
bivector
i = ê1ê2ê3 3
pseudoscalar R
i R R3
i R3
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14. Clifford Algebra Cl3 Non-commutative product w/ A A = A2
Associative: (A B) C = A (B C)
Distributive w.r. to sum of vectors
*symmetric part  dot product
*antisym. part  proportional cross product
Closure: extend the vector space until every product is a linear combination of
elements of the algebra
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15. Subalgebras R Real numbers C = R + iR Complex numbers
Q = R + iR3 Quaternions
Product of two vectors is a quaternion:
represents the oriented surface (plane)
orthogonal to A x B.
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16. Bivector: oriented surface
sweep a b
sweep
antisymmetric, associative
absolute value  area
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17. Differential Calculus
Chain rule:
In 2-d:
Is an “exact differential”?
given
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18. Geometric interpretation:
In 3-d:
Geometric interpretation:
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19. del operator
Examples:
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20. Gradient of r Contour surfaces: spheres  gradient is radial
Algebraically:
and, in general,
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21. Divergence of a Vector Field
E (r)  scalar field (w/ dot product)
It is a measure of how much the filed lines diverge (or converge) from a point (a line, a plane,…)
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22. Divergence as FLUX:
( ) dx
Examples:
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23. Curl of a Vector Field The curl measures circulation about an axis.
Examples:
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24. Clifford product del w/ a cliffor
For a scalar field T = T( r ),
For a vector field E = E( r ),
For a bivector field i B = i B( r ),
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25. Second order derivatives
For a scalar:
For a vector:
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26. What do we mean by “integration”?
cdq
c dr
dw = c(cdq)/ dg = (2pr) dr
| | dl
rdq dr
da= (rdq)dr
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27. Cliffor differentials
dka is a cliffor representing the “volume” element in k dimensions
k =  dl is a vector  e1 dx
(path integral)
k = 2  i nda bivector  e1 dx e2 dy
(surface integral)
k = 3  i dt ps-scalar  e1 dx e2 dy e3 dz
(volume integral)
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28. Fundamental Theorem of Calculus
Particular cases:
Gauss’s theorem
Stokes’ theorem
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29. Delta “function” (distribution)
q step “function” 1 x
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30. Divergence theorem and unit point source apply to
for a sphere of radius e 
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31.  and Displacing the vector r by r‘:
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32. Inverse of Laplacian
To solve so
In short-hand notation:
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33. Orthogonal systems of coordinates
coordinates: (u1, u2, u3 )
orthogonal basis: (e1, e2, e3 )
scale factors: (h1, h2, h3 )
volume:
area
displacement vector:
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34. Scale Factors du1 du2 du3 h1 h2 h3 Cartesian dx dy dz 1 Cylindrical dsdf s
Spherical dr dq r
r sinq
polar (s, f ):
cylindrical (s, f, z ):
spherical (r, q, f ):
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35. Grad:
Div:
Curl:
Laplacian:
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36. Maxwell’s Equations Electro-statics: Magneto-statics: Maxwell:
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37. Formal solution
separates into:
and
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38. Electro-statics
Convolution:
where for point charge.
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39. Superposition of charges
For n charges {dq1, dq2, …, dqn }
continuum limit:
where
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40. linear uniform charge density l (x’) from –L to L
field point @ x = 0, z variable z q
| | x -L L
dq’
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41. with so
Limits:
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42. Gauss’s law Flux of E through a surface S:
volume V enclosed by surface S.
Flux through the closed surface:
choose a “Gaussian surface” (symmetric case)
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43. Charged sphere (uniform density) radius R Gaussian surface:
Examples:
Charged sphere (uniform density) radius R
Gaussian surface:
a) r < R:
b) r > R:
as if all Q is origin
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44. Thin wire: linear uniform density (C/m) Gaussian surface: cylinder
Plane: surface uniform density (C/m2)
Gaussian surface: “pill box” straddling plane
CONSTANT, pointing AWAY from surface (both sides)
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45. Boundary conditions for E
Gaussian box w/ small area D A // surface w/ charge density s
with n pointing away from 1
Equivalently:
Component parallel to surface is continuous
Discontinuity for perp. component = s/e0
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46. Electric Potential (V = J/C)
Voltage
solves Poisson’s equation:
point charge Q at the origin
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47. Potential Difference (voltage)
in terms of E: and
Spherical symmetry: V = V (r)
Potential energy: U = q V (joules)
Equi-potential surfaces: perpendicular to field lines
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48. Example: spherical shell radius R, uniform surface charge density s
Gauss’s law  E(r) = 0 inside (r < R)
For r > R:
and
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49. Example: infinite straight wire, uniform line charge density l
Gauss’s law:
and
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50. Electric-Magnetic materials
conductors
surface charge
boundary conditions
2nd order PDE for V (Laplace)
dielectrics
auxiliary field D (electric displacement field)
non-linear electric media
magnets
auxiliary field H
ferromagnets
non-linear magnetic media
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51. Perfect conductors Charge free to move with no resistance
E = 0 INSIDE the conductor
r = 0 inside the conductor
NET charge resides entirely on the surface
The conductor surface is an EQUIPOTENTIAL
E perpendicular to surface just OUTSIDE
E = s/e0 locally
Irregularly shaped  s is GREATEST where the curvature radius is SMALLEST
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52. Induced charge
Charge held close to a conductor will induce charge displacement due to the attraction (repulsion) of the inducing charge and the mobile charges in the conductor
Example: find charge distribution
1. Inner: conducting sphere,
radius a, net charge 2 Q
2. Outer: conducting thick
shell b < r < c, net charge -Q
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53. Work and Energy Work: for n interacting charges:
continuous distribution r
Energy density:
also electric PRESSURE
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54. Capacitors (vacuum) Capacitor: charge +Q and –Q on each plate
define capacitance C : Q = C DV
with units 1F = 1C /1V E d
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55. charging a capacitor C: move dq at a given time from one plate to the other adding to the q already accumulated
in terms of the field E
 energy density
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56. Laplace’s equation V at a boundary (instead of s) one dimension:
capacitor V0  V (0 to d)
V(x) is the AVERAGE VALUE of V(x-a) and V(x+a)
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57. Average value (for a sphere):
Harmonic function has NO maxima or minima inside. All extrema at the BOUNDARY
Average value (for a sphere):
Proof: 
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58. Method of images
Problem: find V for a point charge q facing an infinite conducting plane
equivalent to
potential: charge density: -q q r q
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59. Laplace’s equation in 2-d
Complex variable z = x + iy  w’(z) well defined
both real and imaginary parts of w fulfill
Laplace’s equation
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60. Examples and equipotentials in 2-d
w(z) = z V(x, y) = y
equipotential lines: y = a (const)
w(z) = z V(x, y) = 2xy, or x2 - y 2
e. l.: xy = a (hyperbolae)
w(z) = ln z V(s, f) = ln s, arctan(y/x)
e. l.: s = exp(a)
w(z) = 1/z = exp(-i f) /s V(s, f) = cosf /s
e. l.: s = cosf /a (circles O)
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61. w(z) = z + 1/z V(s, f) = (s – 1/s) sin f e. l.: s = 1 for a = 0
V(x, y) = finite dipole
w(z) = z + 1/z V(s, f) = (s – 1/s) sin f
e. l.: s = 1 for a = 0
w(x) = exp(z) V(x, y) = exp(x) cos y
Vk(x, y) = exp(k x) cos (k y) & exp(k x) sin (k y)
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62. Separation of variables (polar)
Powers of z:
Solution of
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63. Separation of variables (x, y)
k 2 = separation constant
Eigenvalue equation for X and Y
Construct linear combination, determine coefficients w/ B.C. using orthogonality
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64. Example: solution: B.C.: ii) A = 0; iii) D = 0;
iv) quantization k = 1, 2, 3, … 
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65. orthonormality: i)
solution:
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66. Separation of variables - spherical
Assume axial symmetry, i.e. no f dependence
separation constant l (l +1)
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67. Legendre polynomials change of variable orthogonality
“normalization” Pl (1) = 1
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68. Radial function R(r) = r l or r – l – 1
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69. Given V0(q ) on the surface of a hollow sphere (radius R), find V(r)
Soln: Bl = 0 for r < R, and Al = 0 for r > R
r = R:
using orthogonality:
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70. Uncharged conducting sphere in uniform electric field E = E0 k
V(r = R) = 0:
V  -E0 r cos q r >> R:
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71. Physics BYU

72. Electric Dipole finite dipole as
p = qs is the dipole moment for a finite dipole +q -q r+ r r- q s
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73. Multipole expansion
Legendre polynomials: coefficients of expansion in powers of
where and
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74. Multipole expansion of V: using
with
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75. Electric field of a dipole
Choosing p along the z axis
and
Clifford form:
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76. Polarization P(r) polarization, vector field = = dipole moment/volume.
Compare:
w/ potential due to dipole distribution:
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77. Equivalent “bound charges”
Integration by parts:
so that:
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78. and only if is not uniform surface charge:  -- +
where
and only if is not uniform
surface charge: 
In a dielectric charge does not migrate.
It gets polarized.
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79. Electric Displacement field D(r)
Total charge density: free plus bound
Defining
we get: and
Gauss’s law for D:
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80. Example: Find the electric field E produced by a uniformly polarized sphere of radius R
where
and
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81. Example of Gauss’s law for D:
Long straight wire, uniform line charge l, surrounded by insulation radius a. Find D. 
Outer region, we can calculate the electric field:
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82. permittivity material
Linear Dielectrics
D, E, and P are proportional to each other for linear materials
e 0 -
permittivity space e
permittivity material k
e / e 0
dielectric constant ce
k - 1
susceptibility
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83. Theorem:
In a linear dielectric material with constant susceptibility the bound charge distribution is proportional to the free charge distribution.
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84. Boundary conditions Boundary between two linear dielectrics e 1 n e 2
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85. in terms of the unknown P. Total field: and polarization:
Example: Homogeneous dielectric sphere, radius R in uniform external electric field E0
Solution:
in terms of the unknown P.
Total field: and polarization:
eliminating P:
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86. Energy in dielectrics vacuum energy density: dielectric:
Energy stored in dielectric system:
energy stored in capacitor:
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87. Fringing field in capacitors
l F for fixed charge Q: x where so
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88. Magnetostatics duality in Clifford space
Electric
Field lines: from + to -
E(r): vector field
pos. &neg. charge
r (r): scalar source
V (r): scalar potential
units [E] = V/m
Magnetic
Field lines: closed
i B: bivector field
no magnetic monopoles
J(r): vector source
A (r): vector potential
units [B] = Vs/m2
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89. Lorentz force Point charge q in an external magnetic field B
velocity dependent (but no-friction)
perpendicular to motion: no work done
Example: uniform magnetic field B
define cyclotron (angular)
frequency:
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90. equation of motion: solution: with and radius:
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91. Add uniform electric field E perpendicular to B:
a particular solution is given by:
and
Applications:
cyclotron motion
velocity selector
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92. Physics BYU

93. Charge current densities
current [ I ] = Amp = C/s 
flow of charge per unit time
current density
magnetic force
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94. Relation between I and J
a) consider cylinder, radius R, with uniform steady current I
b) for
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95. Charge conservation current across any closed surface
local (differential) form:
steady currents:
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96. Solving Magnetostatic Equation
Two Maxwell equations rewritten as
formally solved as:
where
Explicitly:
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97. Using the magnetic field is given in terms of the vector source as: For current on a wire (Biot-Savart expression):
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98. Calculate the magnetic field due to a uniform steady current I Trig: Biot-Savart: Integration:
dl’ f s I q in
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99. Force between two // currentsI2 a I1 in f
at wire 2:
attractive force per unit length:
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100. Ampere’s Law using Stokes’ theorem we get over “Amperian loop”.
Symmetric case:
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101. Applications of Ampere’s law
B tangent to Amperian loop Current in x direction
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102. Solenoid: n turns/length current I, magnetic field in z direction
Outer loop:
Straddling loop:
Torus: Amperian loop inside the torus
N = TOTAL number of turns
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103. Vector Potential A Poisson equation: Solution: Disadvantages:
integral diverges
it is a vector field (not scalar)
Advantage: in the same direction
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104. Long thin wire with current I
Examples:
Long thin wire with current I so
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105. Using Stokes’ theorem: magnetic flux Amperian loop FOR A
Example: Find the vector field corresponding to a uniform magnetic field.
Using Stokes’ theorem:
magnetic flux
Amperian loop FOR A
For a solenoid inside, and
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106. Boundary conditions for B
Integrating Maxwell’s equation for B
or, equivalently
is CONTINUOUS
while 1 2
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107. Multipole expansion: magnetic moment
Applying the expansion
to the vector potential for a CLOSED loop:
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108. Using the fundamental theorem in 2-d
The dipole term is:
Using the fundamental theorem in 2-d
and
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109. where the magnetic dipole moment is:
Finally:
where the magnetic dipole moment is:
Curl: m
Area
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110. Magnetic field for a dipole
compared to the electric dipole field:
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111. Magnetism in Matter Paramagnets M // B Diamagnets M // -B
Ferromagnets nonlinear - permanent M
Dipoles
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112. Atomic diamagnetism -e
B add B  m Energy: -e
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113. Magnetization Magnetization = (dipole moment)/volume Vector potential:
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114. Sphere - constant magnetization
sphere radius R - uniform M
with magnetic moment
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115. A useful integral over the sphere
corresponds to electric field w/ constant density: using Gauss’s law: Applications: constant charge density, polarization, and magnetization
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116. Bound current densities:
Auxiliary field H
Defining
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117. Tangential velocity & surface current
An equivalent problem: sphere with uniform surface charge s spinning with constant angular velocity w.
Tangential velocity & surface current
equivalent to uniform M w/bound Kb
inside spinning sphere R w
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118. Ampere’s law for magnetic materials
Experimentally: I & V  H & E
Example: copper rod of radius R carrying a uniform free current I
Amperian loop
radius s:
outside M = 0 and
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119. Boundary conditions for B & H
Integrating Maxwell’s equations:
or, equivalently
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120. Linear Magnetic Materials
Susceptibility and permeability:
diamagnetic ~ -10-5
paramagnetic ~ +10-5
Gadolinium ~ +0.5
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121. Bound surface current: example: solenoid
Theorem:
In a linear homogeneous magnetic material (constant susceptibility) the volume bound current density is proportional to the free current density :
Bound surface current:
example: solenoid
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122. Ferromagnetism Permanent magnetization (no external field necessary)
non-linear relation between M and I
hysteresis loop
dipole orientation in DOMAINS
magnetic domain walls disappear with increasing magnetization
phase transition (Curie point): iron T ~ 770 C
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