notes9 - Magnetic Energy bg Just as there is electric...

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1 Magnetic Energy Just as there is electric energy stored in an electric field E b g there is magnetic energy stored in a magnetic field H b g . Recall the expression for the stored electric energy in a system, Ud vd v E VV =⋅ = ∞∞ z z z z z z 1 2 1 2 DE EE ε . 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 J goes from 0 to J DC . Since this is an electrodynamics problem, the derivation of the stored magnetic energy in a system will not be covered here. However, the final expression is given by v H = z z z z z z 1 2 1 2 BH HH μ . Stored electric energy Stored magnetic energy
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2 Boundary Conditions NORMAL COMPONENT OF THE MAGNETIC FLUX DENSITY Consider a cylindrical surface situated at the interface (boundary) between two different materials. From the divergenceless nature of the magnetic field, the magnetic flux density B= H μ bg satisfies ∇⋅ B=0 and therefore, by the divergence theorem, B S nds zz ⋅= ± 0. Using the surface shown above where Δ h 0 and Δ s is small, the closed surface integral becomes BB B 12 S nn n s n s BBs zz ⋅=⋅ +⋅ =− = ⇒− = ±± ± b g c h b g ΔΔ Δ 0 0 ε 1 ε 2 Δ h Area = Δ s ± n
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3 Thus, the normal component of the magnetic flux density must be continuous across the interface. BB 12 −⋅ = bg ± n 0 TANGENTIAL COMPONENT OF THE MAGNETIC FIELD Consider a rectangular path situated at the interface (boundary) between two different materials. From the non-conservative nature of the magnetic field, ∇ × H=J and therefore, by Stokes's theorem, Hd l= Jn CS ds z zz ⋅⋅ ± . Using the contour shown above where Δ h 0 and Δ w is small, the closed path integral becomes l=H t H t Jn C s ww w z + =⋅ ±± ± . c h c h di b g ΔΔ Δ Thus, ε 1 ε 2 Δ h Δ w ± n ± t C 1 C 2 J s
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4 HHt = Jn 12 −⋅ ⇒−= b g ± ± s tt s HH J n The tangential magnetic field is discontinuous across the boundary by the surface current density in the ± n direction ± ± ± nt n ×= c h . This can be written as ± nHH J ×− = bg s . Notice that J s is equal to zero except on a perfect electric conductor. Thus, for a boundary of any other media (ie. dielectrics), = . Furthermore, the boundary condition governing the magnetic field on a perfect electric conductor states that ± nH J × = 1 s .
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This note was uploaded on 10/25/2011 for the course ECE 2317 taught by Professor Staff during the Fall '08 term at University of Houston.

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notes9 - Magnetic Energy bg Just as there is electric...

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