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Magnetostatics; Faraday's Law; QuasiStatic Fields
Reading: Jackson 5.1 through 5.12, 5.15 through 5.18
Review of basic magnetostatics (i.e., cases with steady currents):
There are no magnetic monopoles.
Magneticflux density (or, magnetic induction) is produced by currents.
Continuity eqn:
In magnetostatics,
is current density: amount of positive charge crossing unit area per
BiotSavart Law: A current I flowing along a differential lengthyields a differential flux densityelement at position (points
from length element to observation point):
1
Force on an element of current:
Torque on a magnetic dipole:
For a distribution of current with density exposed to an external
BiotSavart:
Since
2
Vector identity:
3
Vector identity:
=>
4
in magnetostatics
= 0 for a localized current dist
5
Consider an open surface S bounded by a closed curve C.
Stokes's Thm => (Ampère's Law)
The Vector Potential (Jackson sec 5.4)
We want to solve the eqns of magnetostatics:
If in the region of interest, then we can introduce a magneticscalar potential
M and In this case, we can use the
techniques of electrostatics for solving Laplace's eqn (more later).
More generally,
is the “vector potential”.
6
From Jackson 5.16 (or, p 3 of lecture notes),
[2nd term is OK since
is called a “gauge transformation”.
We can use gauge transformations to yield a convenient form for
Coulomb gauge:
(each Cartesian component of satisfies Poisson's eqn)
7
for all space if the current dist is localized (from slides 4 and 5)
=> = const for Coulomb gauge applied to all space
Analogous to scalar potential for all space:
8
A Circular Current Loop (Jackson sec 5.5)
'
r
x
y
zThe loop has radius a, lies in the xy plane, is centered on the origin, and carries current I.
Spherical coords:
9
Cylindrical symmetry => freedom to place observation point in the xz plane ( = 0)
10
11
Recall electric dipole fields:
=> magnetic field has dipole character, with magnetic dipole moment
12
Next, consider the magnetic induction far from a localized, arbitrarycurrent dist (Jackson sec 5.6).
So, each Cartesian component is
13
To further simplify, we first derive a useful result. Supposeare arbitrary functions and is localized
(but not necessarily divergenceless).
for a localized current dist
Now, take f = 1, g = xi', and impose
14
=>
= 0 = 0
15
Likewise for i = 2, 3
(2)
16
Define the magnetic moment density, or magnetization, as
magnetic moment
Using lots of identities on the cover of Jackson, and with= a unit vector along
17
=3 =0 =0
This has the same form as the field due to an electric dipole:
So, any localized current dist produces a dipole magnetic induction tofirst order at large distances.
18
If the current is restricted to a single plane loop (of arbitrary shape),then
height of triangle
points ⊥ to the plane (use right hand rule with current)
x' = base length
= area of triangle
Note that the circular current loop discussed earlier is a specific case ofthis general result (see slide 12).
19
Force and torque on (and energy of) a localized current distribution inan external magnetic induction (Jackson sec 5.7)
Brief mathematical prelude on the LeviCivita tensor ijk
ijk
= 1 for i, j, k = 1, 2, 3
1 for i, j, k = 1, 3, 2
2, 3, 13, 1, 2
2, 1, 33, 2, 1
0 otherwise (i.e., for 2 or more indices equal)
For example, take i=1:
20
kth component of the induction:
ith component of force:
= 0 for steady, localized current dist (see slide 15)
Eq (2) on slide 16:
21
For example,
(from Jackson cover; explicitly verified on next page)
=>
22
Explicit verification, for i=1:
So, to get the lowestorder force on a current dist due to an external1) Pick an origin within the dist, 2) Compute wrt that origin,3) Take the derivative 4) Evaluate at the origin.
23
Torque:
Since the second integral is
Recall eq 1 on slide 14:
On top of slide 22, we found
to lowest order
24
Potential energy of a permanent magnetic dipole in an external induction:
Also,
25
SP 5.1—5.5
Macroscopic Equations and Boundary Conditions (Jackson sec 5.8)
Magnetic dipole moment of matter comes from 1) current of electronsand 2) intrinsic magnetic moments of atoms.
As with electrostatics, averaging of the microscopic eqns yields themacroscopic eqns:
(only free current is included here)
is called the “magnetic field”.
is the magnetization, i.e., dipole moment per unit volume)
effective current density
For linear, isotropic, paramagnetic and diamagnetic materials,
( is called the “magnetic permeability”)
26
(Jackson p. 192; analogous to electrostatic development in Topic 4, slides 8 through 10)
More complicated for ferromagnetic materials, where dependson history and may be nonzero even for zero applied induction.
Boundary conditions:
= free surface current density
First eqn yields
If 1 ≫
2, then has a much larger normal than tangential
component (as long as is not huge) => is normal to theboundary surface, just as an external electric field is at a conducting surface.
(i.e., current per unit transverse length)
27
Magnetic BoundaryValue Problems when (Jackson sec 5.9)
For linear media =>
If is piecewise constant, then
in each region of constant .
For “hard ferromagnets”, is fixed (doesn't depend on applied field).
where the effective magneticcharge density
28
(magnetic scalar potential)
If there are no boundary surfaces, then in analogy with electrostatics
If is wellbehaved and localized, then integration by parts yields
Far from the region of nonvanishing magnetization,
= magnetic dipole moment
Suppose is discontinuous: inside the ferromagnet andoutside.
29
Gaussian pillbox straddling the surface:
= outwardlydirected normal
“Magnetic charge” inside
=> effective magnetic surfacecharge density
30
SP 5.6
Faraday's Law (Jackson sec 5.15)
If a closed curve C bounds surface S, then
where is the electric field at in its rest frame. The time derivativeis a total derivative, accounting both for time variations in andmotion of the loop in space.
Adopting the rest frame of the loop:
This generalizes for static fields to the dynamic case.
is called the “electromotive force” (emf)
31
Magnetic Field Energy (Jackson sec 5.16)
The work done to generate the currents that yield a static magnetic field is
For a linear medium,
For a localized current distribution,
32
Inductance (Jackson sec 5.17)
Consider N currentcarrying circuits (the ith one has current Ii) in vacuum.
with
33
(“selfinductance”)
(“mutual inductance”)
34
If the circuits are negligibly thin, then
= magnetic flux in i due to field produced by j, divided by current in j
Similarly for selfinductance.
SP 5.7, 5.8
Quasistatic Magnetic Fields in Conductors (Jackson sec 5.18) 35
Suppose the variation in is sufficiently slow that dominatesover the induced
Conductor: ( = conductivity)
Faraday's Law:
In cases of negligible free charge, the variation ofis the only source of
Consider a medium with uniform and frequencyindependent permeability and frequencyindependent conductivity .
Ampère's Law:
Adopt Coulomb gauge:
36
(diffusion eqn)
Taking time derivative:
(diffusion eqn again)
Suppose the field initially varies on a spatial scale ~ L
If the timescale for field decay is , then
Or, if the conductor is subjected to external fields that vary withfrequency = 1, then the fields penetrate into the conductor to a distance
37
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