Applied Engineering Electromagnetics: EE 463 Dr. Zewde 1 : WICHITA STATE UNIVERSITY EE 463: Applied Engineering Electromagnetics Magnetostatics Dr. Zewde 2 Magnetostatics - Static Magnetic Field • Earth's magnetic field is the magnetic field that extends from the Earth’s out into space. • The Earth and most of the planets in the Solar System, as well as the Sun and other stars, all generate magnetic fields through the motion of electrically conducting fluids. Dr. Zewde 3 Basic laws of Magnetostatics • The second law is about the • The first law states that the divergence of curl of magnetic field, i.e., magnetic field at every point (x,y,z) is zero, × = i.e., • Applying Stoke’s Theorem , = 0 ⋅ • Applying Divergence Theorem, d = 0 d = 0 � ⋅ where Ienc is the total current enclosed by the contour C and µ0 is the permeability of � ⋅ The basic laws of magnetostatics specify the divergence and free space. J is the volume current density curl of the static magnetic field B in free space. measured in A/m2. 9 the line integral of magnetic flux intensity around a closed Ampere’s law path is the same as the net current enclosed by the path. The steps of applying Ampere’s law are 1. Recognize the symmetry d = = 2. Sketch the magnetic field lines 0 0 �where ⋅ is the volume� current ⋅ density. 3. Choose a contour C parallel to the field lines, and form a closed contour 4. Solve for the magnetic flux density B. d = = A magnetic field line 0 0 �where ⋅ is the surface � current ⋅ density. is a path to which B is tangential at every point on the line. Dr. Zewde 6 Example: Determine the magnetic field due to the current carrying wire shown in the figure below. Note that J is the uniform volume current density. d = � ⋅ 0 Dr. Zewde 7 Example: Determine the magnetic field due to a) current carrying hollow cylinder b) coaxial current carrying cylinders d = � ⋅ 0 Dr. Zewde 8 Exercise: Problem 6.4: A conducting cylinder of radius R and infinite length carries the nonuniform volume current density J where = ( ) for 0 < < R. Find the magnetic flux density B everywhere. 0 Problem 6.8: A hollow cylinder (tube) of radii a, b and infinite length carries the nonuniform volume current density = ( ) a < < b. Find the magnetic flux density B everywhere. 0 Dr. Zewde 9 The Biot-Savart law • The magnetic fields of arbitrary current distributions can be determined as follow: • For surface current density × = 4 ′ 0 • For filamentary current� I 3 × = 4 ′ 0 � 3 × = (dl’ has the direction of I) 4 ′ 0 � 3 Dr. Zewde 12 Exercise: Determine the magnetic flux density of a current-carrying loop on the z-axis. Field point (0, 0, z) × = 4 ′ 0 � 3 Source point (a, , 0) = + I = −I Dr. Zewde 13 Exercise: Determine the on-axis field of a current-carrying loop considering a partial loop which is part of a closed circuit. × = 4 ′ 0 � 3 ? Dr. Zewde 14.