Finish EM Ch.5: MagnetostaticsMethods of Math. Physics, Friday 1 April 2011, E.J. Zita

• Review & practice

• Lorentz Force

• Ampere’s Law

• Maxwell’s equations

• Magnetic vector potential A || Electrostatic potential V

• Multipole expansion

• Boundary Conditions

Lorentz ForceProblem 5.39 p.247: A current I flows to the right through a rectangular bar

of conducting material, in the presence of a uniform magnetic field B pointing out of the page (Fig.5.56).

(a) If the moving charges are positive, in which direction are they deflected by B? Describe the resultant E field in the bar.

(b) Find the resultant potential difference between the top and bottom of the bar (of thickness t and width w) and v, the speed of the charges. Hall Effect!

Ampere’s Law

p.231 #13-16You choose…

0 ,boundary

B d I I J d l a

Ampere’s Law

p.231 #14

0 ,boundary

B d I I J d l a

Ampere’s Law

p.231 #15

0 ,boundary

B d I I J d l a

Ampere’s Law

p.231 #16

0 ,boundary

B d I I J d l a

Four laws of electromagnetism

Electric Magnetic

Gauss' Law

Charges → E fields

Gauss' Law

No magnetic monopoles

Ampere's Law

Currents → B fields

Faraday's Law

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 equationsElectric Magnetic

Gauss' Law

Charges make E fields

Gauss' Law

What if there were magnetic monopoles?

Ampere's Law

Currents make B fields (so does changing E)

Faraday's Law

Changing B make E fields

Maxwell’s Eqns with magnetic monopole

Lorentz Force:

Continuity equation:

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).

Practice with vector field theorems

Magnetic vector potential

Magnetic vector potential

Electrostatic scalar potential V

Find the vector potential A…

5.27 p.239: Find A above and below a plane surface current flowing over the x-y plane (Ex.5.8 p.226)

B = ____ A || K

ˆKK x

ˆ( )A zA x

ˆ ˆ ˆ

( ) 0 0

x y z

A z

x y z

B A

Electric dipole expansionof an arbitrary charge distribution (r) (p.148)

Pn(cos) are the Legendre Polynomials

Magnetic multipole expansion

Vector potential A of a current loop is (5.83) p.244

Magnetic field of a dipole

B=A, where

ˆ ˆsinm m r φ

03

ˆˆ( ) 2cos sin4dip

m

r

B r A r θ

Find the magnetic dipole moment of a spinning phonograph record of radius R, carrying uniform surface charge ,

spinning at constant angular velocity . (5.35 – see 6.a)

dI = K dr, m = I area =

HW: 5.58 – see 6.b

Boundary conditions

http://resonanceswavesandfields.blogspot.com

See Figs.5.49, 5.50, p.241:

5.31: Check BC for solenoid or spinning shell

(a) Check Eqn. 5.74 for Ex.5.9 (p.277): solenoid.

(b) Check Eqn. 5.75 & 5.76 for Ex. 5.11 (p.236): charged shell

0

0

ˆ(5.74) :

(5.75) :

(5.76) :

above below

above below

above below

n n

B B K n

A A

A AK