Chapter 6 - Chapter 6: Maxwell Equations, Macroscopic...

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6.1 Mawell’s Displacement Current; Maxwell Equations Chapter 6: Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws So far, we have the following set of laws : , , and 0 (6.1) Taking the divergence of , we obtain ρ ∇⋅ = ∇× = ∇× =− ∇⋅ = = t The Displacement Current : B DH J E B HJ 0 (6.2) 0 if 0 This violates the law of conservation of charge. ρρ ∂∂ ∇⋅∇× =∇⋅ = ⇒∇ ⋅ + ±²³ ²´ tt J
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sin Vt ω displacement current in the gap C I real current on the wire 6.1 Mawell’s Displacement Current; Maxwell Equations ( continued ) Maxwell observed that if we postulate , (6.5) where is called the displacement current by Maxwell, then, D t t ∇× = + D D HJ J 0 0, which is consistent with the conservation of charge. (6.5) can be written: , The immediate significance of (6.5) D tt ρ ∂∂ × = ⋅+ = D J J ±²³ ²´ is that it establishes a new mechanism to generate the -field, i.e. by a time-varying -field. : Example of the displacement current BE
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In (6.1), replacing with , we have a new set of equations called the Maxwell equations: 0 ρ ∇× = = + ∇⋅ = =− ∇⋅ = ⎪∇× t t t D B D The Maxwell Equations : HJ B E D (6.6) These 4 equations form the basis of all classical electromagnetic phenomena. As discussed in Ch. 5, Faraday's law connects and . As will be EB shown in Ch. 7, (6.6) lead to EM waves. Thus, Maxwell's theory connects "optics" and "electromagnetism". On the other hand, the Lorentz force equation, , connects "mechanics" and "electromagneti =+ × fE J B sm". 6.1 Mawell’s Displacement Current; Maxwell Equations ( continued ) homogeneous equations inhomogeneous equations
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0 3 3 3 0 0 3 (pp. 27-30) () ( ) 1 4 || ( ) 4 : (b) (a) 0 (c) ρ ε πε μ π ′′ − ′ ×− − ′ ∇⋅ = = ∇× = = Physical laws d d xx x Jx x x Review of Laws & Equations Obtained under Static Conditions : E Ex E Bx 0 0 32 3 0 0 (pp. 178-9) 1 4 4 0( e ) (d) (f) : , , Which of the above laws/equations sti ll hold φφ φ − ′ − ′ ∇× = =−∇ = =− =∇× = Scalar and vector potentials d Questi d on x x x Jx B BJ E BA A true if 0? Why? t 6.1 Mawell’s Displacement Current; Maxwell Equations ( continued )
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6.1 Mawell’s Displacement Current; Maxwell Equations ( continued ) 3 3 1 2 1 2 : (4.89) (5.148) : =⋅ E B Field energy Wd x x Forces ED BH 3 3 21 () (4.40) 0 ( ) 0 (5.86 ρ σ = −⋅ = −× = = E B dx Boundary conditions fE fJ B DD n EE n BB n ) ( ) (5.87) Which of the above equations still hold true if 0? Why : ? ×−= t Question nH H K
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6.2 Vector and Scalar Potentials From the 2 homogeneous Maxwell equations, we may find a vector potential and a scalar potential to represent and . 0 (6.7) φ ∇⋅ = =∇× AE B BB A () 2 0 00 (6.9) With (6.7) and (6.9), we write the 2 inhomogeneous Maxwell equations (for vacuum medum) in terms of and as follows ρε ∂∂ ∇× + = ⇒∇× + = + =−∇ ⇒= ⇒∇ tt t t B EE A E A EA A E 2 22 2 2 11 0 0 0 ( ) (6.10) (6.11) Thus, the set of 4 ρ ε μμ μ +∇ = ∇× = +
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Chapter 6 - Chapter 6: Maxwell Equations, Macroscopic...

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