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FWLec15

# FWLec15 - A F Peterson Notes on Electromagnetic Fields...

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 10/04 Fields & Waves Note #15 Inductance and Magnetic Energy Objectives: Review the equations of magnetostatics and introduce an expression for the energy stored in the magnetic field. Then, introduce the concept of inductance and determine the inductance per unit length of several structures. The Equations of Magnetostatics Magnetostatics involves a distribution of steady (time invariant) electric current and the associated magnetic fields or flux. The primary equations relating these quantities in integral form are Ampere’s Law Hd Jd S CS = ÚÚ Ú l (15.1) and Gauss’ Law Bd S S = ÚÚ 0 (15.2) as described in Note #13. These integral relationships can be used to derive the differential forms of these equations, which are given by —¥ = HJ (15.3) = B 0 (15.4) As discussed previously, (15.3) relates the volume current density to the degree of “twisting” associated with the H -field, while (15.4) dictates that the B -field (and indirectly the H -field) may not spread out or diverge at any point. In addition to these laws, B and H are related by the constitutive relation BH = m (15.5) At an interface between two materials, the fields must satisfy the conditions ˆ () nHH J s ¥- = 12 interface (15.6) ˆ nB B -= 0 interface (15.7)

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 10/04 where ˆ n is the normal vector pointing into region 1, and J s denotes a surface current density (if any) flowing on the interface. Equation (15.7) is equivalent to ˆ () nH H -= mm 11 2 2 0 interface (15.8) A steady distribution of current implies that energy is stored in the system. The energy stored in the magnetic field may be obtained from WB H d vH Hdv m = = ÚÚÚ ÚÚÚ 1 2 1 2 m (15.9) where the integration domain includes any region where the fields are nonzero, and the energy W m is given in units of Joules (J). The integrand in (15.9) is the energy density wH H m = Ê Ë Á ˆ ¯ ˜ 1 2 J m 3 (15.10) The storage of magnetic energy implies the presence of inductance in the system. Inductance Given a current I flowing along some closed path (Figure 1), the inductance L is defined by the expression L II Bd S m S == ÚÚ Y 1 (15.11) where S is an open surface terminating on the contour C of the path that the current follows, and B is the magnetic flux density produced by that current. Inductance has units of Weber/Ampere, or Henry (H). We illustrate the calculation of inductance for several examples. Example: A coaxial cable made of two concentric hollow cylinders separated by a material with permeability supports a total current I that flows in one direction down the center conductor and returns on the outer conductor. Determine the inductance per unit length of the cable. Solution: Figure 2 shows the cable cross section. The inner cylinder, of radius a , supports a surface current density of

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 10/04 J I a z s = 2 p ˆ (15.12) The current returns in the - ˆ z direction on the outer conductor ( r = b ), in the form of a surface current density J I b z s = - 2 ˆ (15.13)
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FWLec15 - A F Peterson Notes on Electromagnetic Fields...

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