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# Chapter 5 - Chapter 5 Magnetostatics Faradays Law...

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d a , ρ J B arbitrary volume 33 We begin with the law of conservation of charge: conservation 0 (5.2) of charge Magnetost vv Q tt t dx d ∂∂ ∇⋅ = =− ⎡⎤ ⇒∇⋅ + = ⎢⎥ ⎣⎦ ∫∫ JJ a J v atics is applicable under the static condition. Hence, 0 and (5.2) gives [for magnetoststics] (5.3) Assuming a magnetic force is experienced by charge moving at v B t q =∇ = J F elocity , we define the magnetic induction by the relation: , which is consistent with the definition in (5.1). B q vB Fv B 5.1 Introduction and Definitions Chapter 5: Magnetostatics, Faraday’s Law, Quasi-Static Fields

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2 I 1 I 1 d A 2 d A x loop 1 loop 2 P 12 5.2 Biot and Savart Law 11 2 1 21 2 2 33 12 12 () || dd d d = ∫∫ xx ±²³ ²´ vv v v AA A A 12 12 0 2 12 1 d d = =− = x x x 2 0 2 2 3 12 The Biot-Savart law states that the differential magnetic field at point (see figure) due to a differential current element in loop 2 is given by 4 μ π × = d Pd d dI B x B x A A 2 0 2 2 3 12 linear superposition, an experimental fact (5.4) Thus, the total field at due to in loop 2 is: (1) 4 Integrating the × ⎡⎤ = ⎢⎥ ⎣⎦ v PI d I x B x A 12 1 2 3 12 12 1 1 0 force on in loop 1 due to in loop 2, we obtain (5.7) 4 ×× = ±²²²³²²²´ v II Id x x FB A 0 1 2 3 12 4 x x
12 1 2 = xx x 1 x 2 x 2 I B 2 d A 0 P 2 3 12 2 2 2 3 00 3 0 21 2 3 12 4 || Rewrite (1): Change to , to , and let , we obtain 1 ( ) ( ) ( ) 44 | | d I Id dad dx μ π μμ ππ × = == ′′ = × x x Gauss Law of Magnetism : B x x J J Bx Jx A v A A 3 3 0 0 3 0 () 1 [ ( )] 4| | | | (5.16) | 0 [Gauss = =∇ × × −− × ⇒∇ = B ±²³ ²´ law of magnetism] (5.17) 5.3 Differential Equations of Magnetostatics and Ampere’s Law x x B 0 3 1 =−∇ operates on x ψ ψψ ∇× =∇ × + ∇× aa a cross section of wire

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da n d A arbitrary loop 23 3 0 0 0 4( ) () 3 || 3 1 0 Rewrite (5.16): ( ) 4| | ( ) 4 4 [] dx πδ μ π −− − ′ ′′ =∇ × ⇒∇× × × xx Jx Ampere's Law : Bx ±²²³ ² ²´ µ²¶²· 0 0 0 (through the loop) , ( 5 . 2 2 ) Ampere's law dI da da = ⇒∇ × = ⇒⋅ = B B x J x Bn Jn B v µ²¶ ²· v A A a much more elaborate representation of the Biot-Savart law (5.25) ⎡⎤ ⎢⎥ 5.3 Differential Equations of Magnetostatics and Ampere’s Law ( continued ) 3 0 3 11 3 1 33 1 0 0 ∇⋅ + =− ⋅∇ ∫∫ ±²³²´ µ²²¶ ² ²· 2 ( ) ∇× ∇× =∇ ∇⋅ aa a
3 0 () || Rewrite (5.16): ( ) 4 , (5.27) where the vector potential is given by dx μ π =∇ × ⇒= × Jx xx Vector Potential : B x BA A 3 0 (5.28) 4 which shows that may be freely transformed (without changing according to (gauge transformat ψ =+ →+ A AB ) AA 0 3 4| | ion) (5.29) We may exploit this freedom by choosing a so that 0 (Coulomb gauge) (5.31) (5.28) d ∇⋅ = ⇒∇⋅ = A A 0 22 2 , Coulomb gauge requires

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Chapter 5 - Chapter 5 Magnetostatics Faradays Law...

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