Chapter 5 Magnetostatics 5.1 The Lorentz Force Law 5.2 The Biot-Savart Law 5.3 The Divergence and Curl of 5.4 Magnetic Vector Potential

5.1.1 Magnetic Fields Charges induce electric field Source charges Test charge

5.1.1

Ex.1 Cyclotron motion relativistic cyclotron frequency EM wave relativistic electron cyclotron maser microwave lightlaser Magnetic Force Lorentz Force Law moment cyclotron frequency

~1960 EM : maser [ 1959 J.Schneider ; A.V. Gaponov] ES : space [1958 R.Q. Twiss ] (1976) K.R. Chu & J.L. Hirshfield : physics in gyrotron/plasma 1978 C.S. Wu & L.C. Lee : EM in space () 1986 K.R. Chen & K.R. Chu : ES in gyrotron relativistic ion cyclotron instability 1993 K.R. Chen ES in Lab. plasma [fusion ( EM ? Lab. & space plasmas ? )] (2)

Ex.2 Cycloid Motion assume (3)

5.1.2 (4)

5.1.2 (5)

Magnetic forces do not work (6)

5.1.3 Currents The current in a wire is the charge per unit time passing a given point. Amperes 1A = 1 C/S The magnetic force on a segment of current-carrying wire

surface current density the current per unit length-perpendicular-to-flow The magnetic force on a surface current is (mobile) (2)

volume current density The current per unit area-perpendicular-to-flow The magnetic force on a volume current is (3)

Ex. 3 Sol. Ex. 4 (a) what is J ? (uniform I) Sol (4)

(b) For J = kr, find the total current in the wire. Sol (5)

relation? Continuity equation (charge conservation) (6)

5.2.1 Steady Currents Stationary charges constant electric field: electrostatics Steady currents constant magnetic field: magnetostatics No time dependence

5.2.2 The Magnetic Field of a Steady Current Biot-Savart Law: for a steady line current Permeability of free space Biot-Savart Law for surface currents Biot-Savart Law for volume currents for a moving point charge

5.2.2 (2) Solution: In the case of an infinite wire,

5.2.2 (3) Force? The field at (2) due to is The force at (2) due to is The force per unit length is

5.2.2 (4) 2

5.3.1 an example: Straight-Line Currents R

5.3.2 The Divergence and Curl of Biot-savart law

for steady current To where Ampere’s law in differential form 5.3.2

5.3.3 Applications of Ampere’s Law Ampere’s Law in differential form Ampere’s Law in integral form Electrostatics: Coulomb Gauss Magnetostatics: Bio-Savart Ampere The standard current configurations which can be handled by Ampere's law: 1.Infinite straight lines 2.Infinite planes 3.Infinite solenoids 4.Toroid

5.3.3 (2) ? Ex.7 symmetry Ex.8

Ex.9 loop 1. loop (3)

Ex.10 Solution: (4)

5.3.3 (5)

5.4.1 The Vector Potential E.S. : M.S. : a constant-like vectorfunction is a vector potential in magnetostatics If there is that, can we find a function to obtain with Gauge transformation

5.4.1 (2) Ampere’s Law if

5.4.1 (3) Example 11 Solution: For surface integration over easier A spiring sphere

5.4.1 (4)

5.4.1 (5)

5.4.1 (6) if R > S if R < S

5.4.1 (7)

5.4.1 (8) Note: is uniform inside the spherical shell

5.4.1 (9) =

5.4.2 Summary and Magnetostaic Boundary Conditions

5.4.2 (2)

5.4.2 (3)

5.4.3 Multipole Expansion of the Vector Potential line current =0 monopole dipole

5.4.3(2)

5.4.3(3) Ex. 13

5.4.3(4) Field of a “pure” magnetic dipoleField of a “physical” magnetic dipole