EM & Vector calculus #5 Physical 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 Ch.1.4 (Curvilinear coordinates): VC4.pdf Ch.1.5 (Dirac Delta): VCdd.pdf Lecture notes at

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

Multipole expansion

Background: vector area

Magnetic Dipole