Finish EM Ch. 5: Magnetostatics Methods of Math Finish EM Ch.5: Magnetostatics Methods 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 Force Problem 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). If the moving charges are positive, in which direction are they deflected by B? Describe the resultant E field in the bar. 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 #13-16 You choose…

Ampere’s Law p.231 #14

Ampere’s Law p.231 #15

Ampere’s Law p.231 #16

Four laws of electromagnetism

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

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 

Electric dipole expansion of an arbitrary charge distribution r(r). (p Pn(cosq) 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

Find the magnetic dipole moment of a spinning phonograph record of radius R, carrying uniform surface charge s, spinning at constant angular velocity w. (5.35 – see 6.a) dI = K dr, m = I area = HW: 5.58 – see 6.b

Boundary conditions See Figs.5.49, 5.50, p.241: http://resonanceswavesandfields.blogspot.com

5.31: Check BC for solenoid or spinning shell Check Eqn. 5.74 for Ex.5.9 (p.277): solenoid. Check Eqn. 5.75 & 5.76 for Ex. 5.11 (p.236): charged shell