Lecture 18 Lecture 18 --The Magnetic Field of a CurrentThe Magnetic Field of a CurrentChapter 33 Chapter 33 --Thursday March 22ndThursday March 22nd•Review of the magnetic field of a current•Use of the Biot-Savartlaw•A reminder about electrostatics and Gauss’ law•Some more vector calculus•Ampère’s lawReading: pages 749 thru 765 (up end of Ch. 33) in HRKReading: pages 749 thru 765 (up end of Ch. 33) in HRKRead and understand the sample problemsRead and understand the sample problemsWebAssign deadline will be Monday 26th at 11:59pmWebAssign deadline will be Monday 26th at 11:59pmHomework set (Ch. 33): E12, E22, E24, E28, E32, E35Homework set (Ch. 33): E12, E22, E24, E28, E32, E35Practice problems from the text:Practice problems from the text:Ch. 33 Ch. 33 --Ex. 27, 33, 37; Prob. 13Ex. 27, 33, 37; Prob. 13121 2 1 22oiFiLBiLdμπ==The magnetic force between two wiresThe magnetic force between two wires21 2 1i=×FLBGGGμochosen so that when i1= i2= 1 A, and L = d = 1 m, F21= 2×10-7N7410 Tm/Aoμπ−=× ⋅2oiBrμπ=The magnetic field due to a wire in 3DThe magnetic field due to a wire in 3DLike Coulomb’s Law:(For an infinite line charge)2oErλπε= or 1 ooqidsvqdqidsμλε→→ = →The magnetic field due to a moving chargeThe magnetic field due to a moving chargeLike Coulomb’s Law:(For a point charge)214oqErπε= or 1 ooqidsvqdqidsμλε→→ = →24ovqBrμπ=22ˆ4sin4ooqrqvBrμπφμπ×==vrBGGThe BiotThe Biot--Savart LawSavart Law23ˆ44ooid iddrrμμππ××==sr srBGGGGThe BiotThe Biot--Savart LawSavart Law23ˆ44oodidridrμπμπ=×=×=∫∫∫BBsrsrGGGGGLetLet’’s think back to electrostaticss think back to electrostaticsencEooqdρεε⎡⎤Φ= ⋅ = ∇⋅ =⎢⎥⎣⎦∫EA EGGGv••This is GaussThis is Gauss’’law, i.e. the more fundamental Maxwell equation.law, i.e. the more fundamental Maxwell equation.••It tells us that EIt tells us that E--fields begin and end on electric charges.fields begin and end on electric charges.••Provides a simple method for calculating E for certain symmetrieProvides a simple method for calculating E for certain symmetries.s.As far as we know, there is no magnetic equivalent of charge.As far as we know, there is no magnetic equivalent of charge.Therefore, magnetic field lines never begin or end.Therefore, magnetic field lines never begin or end.00Bd⎡⎤⇒Φ = ⋅ = ∇⋅ =⎣⎦∫BA BGGGvConsequently, GaussConsequently, Gauss’’law of no use in magnetostatics, law of no use in magnetostatics, since there is nothing with which to equate the flux of since there is nothing with which to equate the flux of BB..By the way.....By the way......... we just derived (wrote down) the 2nd Maxwell equation!.... we just derived (wrote down) the 2nd Maxwell equation!Recall: electrostatic forces are conservativeRecall: electrostatic forces are conservative() () bb baa adq d qVdqVaVb⋅= ⋅=−∇⋅= −⎡⎤⎣⎦∫∫ ∫Fs Es sGGGG G••This allowed us to define a scalar potential This allowed us to define a scalar potential VV..••Also implies..Also implies..0d⋅ =∫EsGGv••Consequently, this integral is not much use in electrostatics.Consequently, this integral is not much use in electrostatics.••It is very important in electrodynamics (MaxwellIt is very important in electrodynamics (Maxwell’’s 4th equation).s 4th equation).••However, this is because its However, this is because its not equal to zeronot equal to zeroin electrodynamics.in electrodynamics.MaxwellMaxwell’’s 3rd equation (a.k.a. Amps 3rd equation (a.k.a. Ampèèrere’’s Law)s Law)()12oenc odi iiμμ⋅ ==−∫BsGGvcos ;dBdsθ⋅ =∫∫BsGGvvenc jSid= ⋅ = Φ∫jAGGRightRight--handhand--ruleruleMaxwellMaxwell’’s 3rd equation (a.k.a. Amps 3rd equation (a.k.a. Ampèèrere’’s Law)s Law)The fundamental theorem of calculus:The fundamental theorem of calculus:() ()2121xxdfdx fx fxdx⎛⎞=−⎜⎟⎝⎠∫() ()2121fd f f∇⋅ = −∫rrrr rGGGGGThis gives us two new integral theorems known as the ‘divergence theorem’ (sometimes called Gauss’ law) and the ‘curl theorem’ (also known as Stokes’ theorem).()volume surfacedV d∇⋅=⋅∫∫EEAGGGv()surface boundarylinedd∇×⋅ = ⋅∫∫EA ElGGGGvGauss:Stokes:MaxwellMaxwell’’s 3rd equation (a.k.a. Amps 3rd equation (a.k.a. Ampèèrere’’s Law)s Law)()oSSdddμ⋅ = ∇× ⋅ = ⋅∫∫ ∫Bs B A jAGGGGGGvThe fundamental theorem of calculus:The fundamental theorem of
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