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## Chapter 5 Magnetostatics

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**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**Source charges Charges induce electric field Test charge**EM wave**light laser 5.1.2 Magnetic Force Lorentz Force Law Ex.1 Cyclotron motion moment cyclotron frequency relativistic cyclotron frequency microwave relativistic electron cyclotron maser**5.1.2 (2)**~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 ? )]**5.1.2 (3)**Ex.2 Cycloid Motion assume**5.1.2 (6)**Magnetic forces do not work**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**5.1.3 (2)**surface current density the current per unit length-perpendicular-to-flow (mobile) The magnetic force on a surface current is**5.1.3 (3)**volume current density The current per unit area-perpendicular-to-flow The magnetic force on a volume current is**5.1.3 (4)**Ex. 3 Sol. Ex. 4 (a) what is J ? (uniform I) Sol.**5.1.3 (5)**(b) For J = kr, find the total current in the wire. Sol.**5.1.3 (6)**relation? Continuity equation (charge conservation)**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)**The field at (2) due to is The force at (2) due to is Force? The force per unit length is**5.3.2 The Divergence and Curl of**Biot-savart law**5.3.2**for steady current To where Ampere’s law in differential form**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: • Infinite straight lines • Infinite planes • Infinite solenoids • Toroid**5.3.3 (2)**? Ex.7 symmetry Ex.8**5.3.3 (3)**Ex.9 loop 1. loop 2.**5.3.3 (4)**Ex.10 Solution:**M.S. :**a constant-like vector function 5.4.1 The Vector Potential E.S. : Gauge transformation is a vector potential in magnetostatics If there is that , can we find a function to obtain with**5.4.1 (2)**Ampere’s Law if**5.4.1 (3)**Example 11 A spiring sphere Solution: For surface integration over easier**5.4.1 (6)**if R > S if R < S**5.4.1 (8)**Note: is uniform inside the spherical shell**5.4.2 Summary and Magnetostaic**Boundary Conditions**5.4.3 Multipole Expansion of the Vector Potential**line current =0 monopole dipole**5.4.3(3)**Ex. 13**5.4.3(4)**Field of a “pure” magnetic dipole Field of a “physical” magnetic dipole