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