FWLec13 - A. F. Peterson: Notes on Electromagnetic Fields...

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 10/04 Fields & Waves Note #13 Introduction to Magnetostatics Objectives: Discuss the concepts of magnetic force, magnetic field, and magnetic flux. Present the Biot-Savart Law, Gauss’ Law for the magnetic field, and Ampere’s Circuital Law. Introduce permeability and relative permeability. Finally, obtain the magnetostatic boundary conditions at the interface between two materials. Ampere’s Force Law In a manner similar to the phenomenon of the force experienced between two point charges, two current-carrying wires in empty space (Figure 1) experience a force that can be calculated from the expression FI d Id R R ¥ Ú Ú 22 01 1 2 4 l l m p ˆ curve 1 curve 2 (13.1) Equation (13.1) involves two integrals over the mathematical contours of the wires, where the displacement vector R points from the point of integration on contour 1 to the point of integration on contour 2, and yields the collective force on wire 2. The expression contains a parameter mp 0 7 41 0 = N A H m 2 (13.2) known as the permeability of free space. This parameter plays the role of a constant of proportionality in (13.1), and converts the units accordingly. A Newton per (Ampere) 2 is the same as one Henry per meter. Biot-Savart Law The expression in (13.1) is rather complicated, and seldom produces closed-form expressions for force in practical situations. Instead, we seek an alternative normalization of the force that normalizes out the effect of the second wire. Two alternative fields that can in some sense substitute for the force are the magnetic field H and the magnetic flux density B . The magnetic flux density is the normalized force per Ampere-meter, or B R R = ¥ Ê Ë Á ˆ ¯ ˜ Ú 2 4 l ˆ curve 1 N A-m (13.3) The units of N/(Am) are equivalent to Webers/m 2 (Wb/m 2 ) and to Tesla (T):
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A. F. Peterson: Notes on Electromagnetic Fields & Waves 10/04 N A-m Wb m T 2 == (13.4) The expression in (13.3) removes the effect of the second current-carrying wire in (13.1). A comparable normalization is given by Hxyz Id R R (,,) ˆ = ¥ Ê Ë Á ˆ ¯ ˜ Ú 11 2 4 l p curve 1 A m (13.5) which differs from (13.5) only by the absence of the parameter m 0 . The magnetic field has units of N A N A m 2 ¥= (13.6) It follows that in empty space, the magnetic field and flux density are related by BH = 0 (13.7) It is important to note that neither B nor H point in the same direction as the original force in (13.1). Equations (13.3) and (13.5) are somewhat easier to work with than (13.1), and either equation is known as the Biot-Savart Law . We illustrate the application of the Biot- Savart Law for one example, the infinite straight wire. Example: A thin wire carrying total current I resides along the z -axis. Find the magnetic field produced by this current. Solution: Figure 2 shows the geometry. Cylindrical symmetry is suggested, and since changes in f or z should not affect the result, we locate the observer at the point ( r , 0, 0). The integration along the z -axis involves a differential length vector dd z z l = ¢ ˆ (13.8) The displacement vector pointing from the point of integration on the z -axis (0, 0, z ¢
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FWLec13 - A. F. Peterson: Notes on Electromagnetic Fields...

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