```Finish EM Ch.5: Magnetostatics
Methods of Math. Physics, Thus. 10 March 2011, E.J. Zita
• Lorentz Force
• Ampere’s Law
• Maxwell’s equations (d/dt=0)
• Preview: Full Maxwell Equations (d/dt≠0)
• Magnetic vector potential A || Electrostatic potential V
• BC
• Multipole expansion
Lorentz Force F  qv  B  IL  B
p.220 #10 (a)
Lorentz Force F  qv  B  IL  B
p.220 #12
Ampere’s Law

boundary
B  dl   0 I , I   J d a
p.231 #13-16
You choose…
Ampere’s Law

boundary
p.231 #14
B  dl   0 I , I   J d a
Ampere’s Law

boundary
p.231 #15
B  dl   0 I , I   J d a
Ampere’s Law

boundary
p.231 #16
B  dl   0 I , I   J d a
Four laws of electromagnetism
Electric
Magnetic
Gauss' Law
Gauss' Law
Charges → E fields
No magnetic monopoles
Ampere's Law
Currents → B fields
Electrodynamics
• Changing E(t) make B(x)
• Changing B(t) make E(x)
• Wave equations for E and B
• Electromagnetic waves
• Motors and generators
• Dynamic Sun
Full Maxwell’s equations
Electric
Magnetic
Gauss' Law
Gauss' Law
Charges make E fields
What if there were magnetic
monopoles?
Ampere's Law
Currents make B fields
(so does changing E)
Changing B make E fields
Maxwell’s Eqns with magnetic monopole
Lorentz Force:
Continuity equation:
Vector Fields: Helmholtz Theorem
For some vector field F, if the
divergence = D =  F, and the
curl = C = F
then (a) what do you know about  C ?
and (b) Can you find F?
(a)  C = 0, because  (F)  0
(b) We can find F iff we have boundary conditions, and require the
field to vanish at infinity.
Helmholtz: A vector field is uniquely determined by its div and curl
(with BC)
Vector Fields: Potentials.1
For some vector field F = -  V, find F :
(hint: look at identities inside front cover)
F = 0 → F = -V
Curl-free fields can be written as the gradient of a scalar potential
(physically, these are conservative fields, e.g. gravity or electrostatic).
Theorem 1 – examples
The second part of each question illustrates Theorem 2,
which follows…
Vector Fields: Potentials.2
For some vector field F = A , find F :
F = 0
→ F = A
Divergence-free fields can be written as the curl of a vector potential
(physically, these have closed field lines, e.g. magnetic).
Optional – Proof of Thm.2
Practice with vector field theorems
Magnetic vector potential
Magnetic vector potential
Electrostatic scalar potential V
Electric dipole expansion
of an arbitrary charge distribution r(r)
Pn(cosq) are the Legendre Polynomials
(p.148)
Multipole expansion
Magnetic field of a dipole
B=A, where
ˆ
m  rˆ  m sinq φ
Spring quarter in E&M
•
•
•
•
•
•
•
•
Dynamics! dE/dt and dB/dt
Bohm-Aharanov effect (A >B)
Magnetic monopole: Blas Cabrera’s measurement
Conservation laws, EM energy
EM waves
Relativistic phenomena
EM field tensor
Spring quarter in MMP
• Tuesday: Boas and DiffEq
→ Hamiltonians / Lagrangians
• Thursday: Quantum Mechanics
• Friday: Electromagnetism and Research
```