The Steady State Magnetic Field - Electromagnetics - Lecture Slides, Slides of Electromagnetism and Electromagnetic Fields Theory

Download The Steady State Magnetic Field - Electromagnetics - Lecture Slides and more Slides Electromagnetism and Electromagnetic Fields Theory in PDF only on Docsity! 1 The Steady State Magnetic Field The Concept of Field (Physical Basis ?) Why do Forces Must Exist? Magnetic Field – Requires Current Distribution Effect on other Currents – next chapter Free-space Conditions Magnetic Field - Relation to its source – more complicated Accept Laws on “faith alone” – later proof (difficult) Do we need faith also after the proof? 2 Magnetic Field Sources Magnetic fields are produced by electric currents, which can be macroscopic currents in wires, or microscopic currents associated with electrons in atomic orbits. 5 I NK    d The total current I within a transverse Width b, in which there is a uniform surface current density K, is Kb. For a non-uniform surface current density, integration is necessary. Alternate Forms H S K_x 4  R 2       d aR H v J_x 4  R 2       d aR Biot-Savart Law B-S Law expressed in terms of distributed sources 6 H2   z1 I 4   2 z1 2   3 2         d az  a z1 az  I 4    z1   2 z1 2   3 2        d a H2 I 2   a The magnitude of the field is not a function of phi or z and it varies inversely proportional as the distance from the filament. The direction is of the magnetic field intensity vector is circumferential. Biot-Savart Law 7 Biot-Savart Law H I 4   sin  2  sin  1   a  10 Ampere’s Circuital Law Ampere’s Circuital Law states that the line integral of H about any closed path is exactly equal to the direct current enclosed by the path. We define positive current as flowing in the direction of the advance of a right-handed screw turned in the direction in which the closed path is traversed. LH_dot_    d I 11 Ampere’s Circuital Law - Example LH_dot_    d 0 2  H     d H  0 2  1    d I H I 2   12 H I 2   a  b  a H 0  cH I  2  a 2  2   H I I  2 b 2  c 2 b 2           H I 2   2 b 2  c 2 b 2   H   I 2   c 2  2  c 2 b 2   Ampere’s Circuital Law - Example Ampere’s Circuital Law - Example =z axis K=K, a_atp =py—a,z=0 H=K, mo ay, (inside toroid) H=0 (outside) (a) = axis H= amp ay (well inside toroid) (6) 15 16 CURL The curl of a vector function is the vector product of the del operator with a vector function CURL H- Ho = Ag at Ayo ay + Ho a; 4 3 Ax p H_dot_ dL (curl_H)ay = lim ————— ASy > 0 ASN 20 CURL az  DelXHz 75DelXHz x H1y x y z( ) d d y H1x x y z( ) d d  ay  DelXHy 98DelXHy z H1x x y z( ) d d x H1z x y z( ) d d  ax  DelXHx 420DelXHx y H1z x y z( ) d d z H1y x y z( ) d d  z 3y 2x 5Determine J at: H1z x y z( ) 4 x 2  y 2 H1y x y z( ) y 2  x zH1x x y x( ) y x x 2 y 2   In a certain conducting region, H is defined by: Example 1 21 CURL Example 2 H2x x y x( ) 0 H2y x y z( ) x 2 z H2z x y z( ) y 2  x x 2 y 3 z 4 DelXHx y H2z x y z( ) d d z H2y x y z( ) d d  DelXHx 16 ax  DelXHy z H2x x y z( ) d d x H2z x y z( ) d d  DelXHy 9 ay  DelXHz x H2y x y z( ) d d y H2x x y z( ) d d  DelXHz 16 az  22 Example 8.2 25 Magnetic Flux and Magnetic Flux Density B 0 H 0 4  10 7  H m permeabil i ty in free space  SB_dot_    d SB_dot_    d 0 26 The Scalar and Vector Magnetic Potentials H Del_Vm J 0 Vm a b LH_dot_    d The Scalar and Vector Magnetic Potentials IdL=Idza, 27