notes8 - Magnetostatic Fields Version 1, 11/10/98 Chapt....

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1 Magnetostatic Fields Version 1, 11/10/98 Chapt. 13, Shen and Kong We turn our attention now to the study of magnetostatic fields. The sources for magnetostatic fields are direct currents, J . Since the source is a vector for magnetostatic fields, one might expect magnetic fields to depend quite differently on their sources than do electric fields on their scalar sources. Though this does turn out to be the case, there are nevertheless striking similarities between the two cases, as we shall see. Magnetostatic fields are determined by the two equations ∇× = ∇⋅ = HJ B Ampere's law Magnetic form of Gauss's law bg , . 0 Note the following comparisons with the static forms of Faraday’s and Gauss’s laws: In contrast to Faraday’s law, Ampere’s law has a vector source term, J , a current density. The quantity H [A/m] appearing in Ampere’s law is the magnetic field intensity . In contrast to Gauss’s law for electric fields, the magnetic form of Gauss’s law has no source term. Since its divergence is zero, the flux lines of the magnetic flux density B [W/m 2 ] must either form a closed path or end at infinity. Note that since ∇⋅∇× = = ∇⋅ 0 , the current in Ampere’s law must be a DC current forming a closed circuit. Corresponding to these point-form equations are the integral forms of these equations, obtained by integrating Ampere’s and Gauss’s laws over a surface and volume and using Stokes’s theorem and the divergence theorem, respectively: s ⋅= zz dd CS A Integral form of Ampere's law b g
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2 Bs ⋅= z d S 0 Integral form of magnetic Gauss's law b g In free space, BH and are related by the constitutive equation = μ 0 , where μπ 0 7 41 0 [H/m] is the permeability of free space. The total force (electric and magnetic) on a point charge q moving with velocity v in the presence of the electric field intensity E and the magnetic flux density B is given by the Lorentz force law: FE v B =+ × q bg . Determining Magnetic Fields by Ampere’s Law Similar to the use of Gauss’s law in certain situations with a high degree of symmetry, Ampere’s law alone can be used to determine magnetic field in situations where certain symmetries are present. The fact that current sources are vector quantities and hence have a direction associated with them means that some symmetries (e.g., spherical) will not be allowed. The first case examined is the magnetic field of a line source of current. Magnetic Field Due to a Current Line Source: Consider an infinitely long line source carrying a direct current I along the z -axis, as shown in the figure. Since the source is invariant with respect to rotation about its axis (rotation in φ ) and with respect to translation along the z -axis, a cylindrical coordinate system seems appropriate. Let us apply Ampere’s law along a circular path of radius ρ and centered along the z -axis as I H
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3 shown. Along the circular path, only the H φ component of the magnetic field will contribute to the line integral, and, by symmetry, H must be constant on the path.
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notes8 - Magnetostatic Fields Version 1, 11/10/98 Chapt....

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