em & vector calculus #5 physical systems, tuesday 27 feb. 2007, ejz vector calculus 1.6: theory...
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EM & Vector calculus #5Physical Systems, Tuesday 27 Feb. 2007, EJZ
Vector Calculus 1.6: Theory of vector fields
• Quick homework Q&A
• thanks to David for Dirac Delta during jury duty last week
• Helmholtz Theorem and Potentials
E&M Ch.5.3-4: finishing Magnetostatics
• Quick homework Q&A
• Review, Div and curl of B
• Magnetostatic BC
• Magnetic vector potential
• Multipole expansion of vector potential?
Vector calculus HW
Online solutions at http://192.211.16.13/curricular/physys/0607/solns/
Ch.1.4 (Curvilinear coordinates): VC4.pdf
Ch.1.5 (Dirac Delta): VCdd.pdf
Lecture notes at
http://192.211.16.13/curricular/physys/0607/lectures/
Vector Fields: Helmholtz Theorem
For some vector field F, if the
divergence = D = F, and the
curl = C = F, 0
then (a) what do you know about C ?
and (b) Can you find F?
( )a C = 0, because (F) 0
(b) Can find F iff we have boundary conditions, and require field to vanish at infinity.
Helmholtz: 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
E&M Ch.5b: Magnetostatics
• Quick homework Q&A
• Review, Div and curl of B
• Magnetic vector potential
• Magnetostatic BC
• Multipole expansion of vector potential
Magnetostatic BC
Magnetic vector potential
Magnetic vector potential
Multipole expansion
Background: vector area
Magnetic Dipole
Magnetic Dipole