NOTES9 - 12/01/03 Magnetic Energy Just as electric energy...

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1 2 / 0 1 / 0 3 Magnetic Energy Just as electric energy is stored in an electric field ( ) E , magnetic energy is stored in a magnetic field ( . Recall the expression for the stored electric energy in a system, ) H 11 22 E VV Ud Vd V ε ∞∞ =⋅= ⋅ ∫∫ DE EE . In order to derive an expression for the stored magnetic energy, it is first necessary to calculate the energy required by a source to establish the magnetic field. Stated another way, it is necessary to determine the energy supplied by a source as the current density increases from 0 to its final DC value, . Since this is in reality an electrodynamics problem, the derivation of the magnetic energy stored in a system will not be covered here. However, the final expression is given by J . H V µ =⋅= BH HH Stored magnetic energy (Units of Joules [J]) Stored electric energy (Units of Joules [J]) Boundary Conditions NORMAL COMPONENT OF THE MAGNETIC FLUX DENSITY Consider a cylindrical surface situated at the interface between two magnetic materials of differing permeabilities. Area = s Since the µ 2 µ 1 ˆ n ± n h ˆ n
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magnetic flux density µ = BH satisfies =0 B , then by the divergence theorem, ˆ 0 S dS = Bn v . Using the surface shown above where h 0 and s is sufficiently small that the magnetic field is essentially constant on s , the closed surface integral becomes ( ) ( ) ( ) () 12 ˆˆ ˆ 0 0 S nn dS s s BB s BB ⋅= +⋅ =−∆ = ⇒− = B n B n v Thus, the normal component of the magnetic flux density must be continuous across the interface: ( ) ˆ 0 BBn . TANGENTIAL COMPONENT OF THE MAGNETIC FIELD Consider a rectangular path situated at the interface (boundary) between two different materials. J s ± t ± n ˆ n C 2 C 1 h ε 2 ε 1 w From the non-conservative nature of the magnetic field,
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∇ × H=J , we have, integrating over an open surface with bounding contour C and using Stokes's theorem, S 12 ˆ d= CC S dS + ⋅⋅ ∫∫ HJ n v A . Using the contour shown above where 0 h and w is sufficiently small that the magnetic field is essentially constant on w , the closed path integral becomes ( ) ( ) ( ) () ˆˆ ˆ . s ww w + + =⋅∆ HH t H t Jn v A Thus, ( ) ˆ ˆ = tt s sn HH J ⇒−= HHt Jn The tangential magnetic field is discontinuous across the boundary by the surface current density in the direction nn . Since ˆ ˆ t = × tn , this can be written as n ( ) ( ) ˆ ˆ ˆ == s ′′ ×⋅ − ⋅× − ⎡⎤ ⎣⎦ nnHH nnHH nJ or since the direction of is arbitrary while ˆ n s J and the bracketed quantity are tangential to the surface, 2 A ˆ m s ×− = nHH J . Notice that is equal to zero except on a perfect electric conductor. Thus, for a boundary between any other media (ie. good conductors, dielectrics), J s = .
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NOTES9 - 12/01/03 Magnetic Energy Just as electric energy...

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