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- 1. 1 Chapter 8 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. 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.
- 3. 3 From GSU Webpage Magnetic Field – Concepts, Interactions and Applications
- 4. 4 Biot-Savart Law dH IdL aR 4 R 2 IdL R 4 R 3 Magnetic Field Intensity A/m H L I 4 R 2 d a R Verified experimentally At any point P the magnitude of the magnetic field intensity produced by a differential element is proportional to the product of the current, the magnitude of the differential length, and the sine of the angle lying between the filament and a line connecting the filament to the point P at which the field is desired; also, the magnitude of the field is inversely proportional to the square of the distance from the filament to the point P. The constant of proportionality is 1/4 Biot-Savart = Ampere’s law for the current element.
- 5. 5 I N K 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. 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. 7 Biot-Savart Law H I 4 sin 2 sin 1 a
- 8. 8 Example 8.1 H2x 8 4 0.3 ( ) sin 53.1 180 1 a H2x 8 4 0.3 ( ) sin 53.1 180 1 H2x 3.819 a must be refered to the x axis - w hich becomes az H2y 8 4 0.4 ( ) 1 sin 36.9 180 az H2y 8 4 0.4 ( ) 1 sin 36.9 180 H2y 2.547 H2 H2x H2y H2 6.366 az 1x 90 180 2x atan 0.4 0.3 1y atan 0.3 0.4 2y 90 180
- 9. 9 Ampere’s Circuital Law The magnetic field in space around an electric current is proportional to the electric current which serves as its source, just as the electric field in space is proportional to the charge which serves as its source.
- 10. 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. L H_dot_ d I
- 11. 11 Ampere’s Circuital Law - Example L H_dot_ d 0 2 H d H 0 2 1 d I H I 2
- 12. 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
- 13. 13 Ampere’s Circuital Law - Example
- 14. 14 Ampere’s Circuital Law - Example
- 15. 15 Ampere’s Circuital Law - Example
- 16. 16 CURL The curl of a vector function is the vector product of the del operator with a vector function
- 18. 18 CURL Ampere’s Circuital Law Second Equation of Maxwell Third Equation curl_H = x H x H = J x E = 0
- 19. 19 Illustration of Curl Calculation CurlH y Hz d d z Hy d d ax z Hx d d x Hz d d ay x Hy d d y Hx d d az CurlH ax x d d Hx ay y d d Hy az z d d Hz CURL
- 20. 20 CURL az DelXHz 75 DelXHz x H1y x y z ( ) d d y H1xx y z ( ) d d ay DelXHy 98 DelXHy z H1xx 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 H1xx y x ( ) y x x 2 y 2 In a certain conducting region, H is defined by: Example 1
- 21. 21 CURL Example 2 H2xx 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 H2xx y z ( ) d d x H2z x y z ( ) d d DelXHy 9 ay DelXHz x H2y x y z ( ) d d y H2xx y z ( ) d d DelXHz 16 az
- 22. 22 Example 8.2
- 23. 23 Stokes’ Theorem L H_dot_ d S Del H ( )_dot_ d S The sum of the closed line integrals about the perimeter of every Delta S is the same as the closed line integral about the perimeter of S because of cancellation on every path.
- 24. 24 Example 8.3 Hr r 6 r sin H r 0 H r 18 r sin cos segment 1 r 4 0 0.1 0 segment 2 r 4 0.1 0 0.3 segment 3 r 4 0 0.1 0.3 dL dr ar r d a r sin d a First tem = 0 on all segments (dr = 0) Second term = 0 on segment 2 ( constant) Third term = 0 on segments 1 and 3 ( = 0 or constant) L H d H r d H r sin d H r d since H=0 0 0.3 H r r sin d 22.249
- 25. 25 Magnetic Flux and Magnetic Flux Density B 0 H 0 4 10 7 H m permeability in free space S B_dot_ d S B_dot_ d 0
- 26. 26 The Scalar and Vector Magnetic Potentials H Del_Vm J 0 Vm a b L H_dot_ d
- 27. 27 The Scalar and Vector Magnetic Potentials
- 28. 28 Derivation of the Steady-Magnetic-Fields Laws