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 Theorem𝑜𝑜 , = 0 𝛻𝛻 𝑩𝑩 𝜇𝜇 𝐽𝐽

𝛻𝛻 ⋅ 𝑩𝑩 • Applying Divergence Theorem, d =

0 𝑒𝑒𝑒𝑒𝑒𝑒 d = 0 � 𝑩𝑩 ⋅ 𝒍𝒍 𝜇𝜇 𝐼𝐼 where Ienc is 𝐶𝐶the total current enclosed by

the contour C and µ0 is the permeability of �𝑩𝑩 ⋅ 𝑺𝑺 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 𝒗𝒗 0 𝑒𝑒𝑒𝑒𝑒𝑒 �where𝑩𝑩 ⋅ 𝒍𝒍is the 𝜇𝜇volume� current𝑱𝑱 ⋅ 𝑑𝑑 density.𝑺𝑺 𝜇𝜇 𝐼𝐼 𝐶𝐶 3. Choose a contour C parallel to the field lines, and form a𝑱𝑱𝒗𝒗 closed contour 4. Solve for the magnetic flux density B.

d = =

A magnetic field line 0 𝒔𝒔 0 𝑒𝑒𝑒𝑒𝑒𝑒 �where𝑩𝑩 ⋅ is𝒍𝒍 the surface𝜇𝜇 � current𝑱𝑱 ⋅ 𝑑𝑑 density.𝒍𝒍 𝜇𝜇 𝐼𝐼 is a path to which B 𝐶𝐶 is tangential at every 𝑱𝑱𝒔𝒔 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 =

� 𝑩𝑩 ⋅ 𝒍𝒍 𝜇𝜇0𝐼𝐼𝑒𝑒𝑒𝑒𝑒𝑒 Dr. Zewde 𝐶𝐶 7 Example: Determine the magnetic field due to a) current carrying hollow cylinder b) coaxial current carrying cylinders

d =

� 𝑩𝑩 ⋅ 𝒍𝒍 𝜇𝜇0𝐼𝐼𝑒𝑒𝑒𝑒𝑒𝑒 𝐶𝐶 Dr. Zewde 8 Exercise:

Problem 6.4: A conducting cylinder of radius R and infinite length carries the nonuniform volume current density J where = ( ) 𝜌𝜌 for 0 < < R. Find the magnetic flux density B everywhere. 𝐽𝐽 𝑎𝑎𝑧𝑧 𝐽𝐽0 𝑅𝑅 Problem𝜌𝜌 6.8: A hollow cylinder (tube) of radii a, b and infinite length carries the nonuniform volume current density = ( ) a < < b. 𝑎𝑎 Find the magnetic flux density B everywhere. 𝐽𝐽 𝑎𝑎𝑧𝑧 𝐽𝐽0 𝜌𝜌 𝜌𝜌

Dr. Zewde 9 The Biot-Savart law

• The magnetic fields of arbitrary current distributions can be determined as follow:

• For surface current density × 𝑱𝑱𝒔𝒔 = 4 ′ 𝜇𝜇0 𝑱𝑱𝒔𝒔 𝑹𝑹𝑑𝑑𝑑𝑑 • For filamentary𝐵𝐵 current� I 3 𝜋𝜋 𝑅𝑅 × = 4 ′ 𝜇𝜇0 𝑰𝑰 𝑹𝑹𝑑𝑑𝑑𝑑 𝐵𝐵 � 3 𝜋𝜋 𝑅𝑅× = (dl’ has the direction of I) 4 ′ 𝜇𝜇0𝐼𝐼 𝒅𝒅𝒅𝒅 𝑹𝑹 𝐵𝐵 � 3 𝜋𝜋 𝑅𝑅 Dr. Zewde 12 Exercise: Determine the magnetic flux density of a current-carrying loop on the z-axis.

Field point (0, 0, z) × = 4 ′ 𝜇𝜇0 𝑰𝑰 𝑹𝑹𝑑𝑑𝑑𝑑 𝐵𝐵 � 3 𝜋𝜋 𝑅𝑅

Source point (a, , 0)

𝜙𝜙 = +

𝑹𝑹I = −𝑎𝑎I 𝜌𝜌𝑎𝑎 𝑎𝑎𝑧𝑧𝑧𝑧 𝑎𝑎𝜙𝜙 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 ′ 𝜇𝜇0 𝑰𝑰 𝑹𝑹𝑑𝑑𝑑𝑑 𝐵𝐵 � 3 𝜋𝜋 𝑅𝑅

? Dr. Zewde 14