1. Lecture 8 MAGNETOSTATICS Magnetic Fields Fundamental Postulates of Magnetostatics in Free Space Prof. Viviana Vladutescu

2. Magnetic Fields

3. Magnetism and electricity have not been considered distinct phenomena until Hans Christian Oersted conducted an experiment that showed a compass deflecting in proximity to a current carrying wire

4. Produced by -time varying electric fields -permanent magnet (arises from quantum mechanical electron spin/ can be considered charge in motion=current ) -steady electric currents

5. If we place a wire with current I in the presence of a magnetic field, the charges in the conductor experience another force F m

6. F m ~ q, u, B q –charge u –velocity vector B -strength of the field (magnetic flux density) μ r –relative permeability μ –absolute permeability μ 0 -permeability of the free space M=χ m H- is the magnetization for linear and homogeneous medium

7. Relative permeabilities for a variety of materials Materialμ/(H m -1 )μrμr Application Ferrite U 601.00E-058UHF chokes Ferrite M339.42E-04750Resonant circuit RM cores Nickel (99% pure)7.54E-04600- Ferrite N413.77E-033000Power circuits Iron (99.8% pure)6.28E-035000- Ferrite T381.26E-0210000Broadband transformers Silicon GO steel5.03E-0240000Dynamos, mains transformers supermalloy1.261000000Recording heads

8. Lorentz’s Force equation Note: Magnetic force is zero for q moving in the direction of the magnetic field (sin0=0)

9. When electric current is passed through a magnetic field a force is exerted on the wire normal to both the magnetic field and the current direction. This force is actually acting on the individual charges moving in the conductor.

10. The magnetic force is exerting a torque on the current carrying coil

11. Cross product

12. Fundamental Postulates of Magnetostatics in Free Space

15. Law of conservation of magnetic flux

16. There are no magnetic flow sources, and the magnetic flux lines always close upon themselves

17. Ampere’s circuital law The circulation of the magnetic flux density in free space around any closed path is equal to μ 0 times the total current flowing through the surface bounded by the path

18. Stoke’s Theorem Note: For a closed surface there will be no surface bounding external contour

19. Proof: Sum over where Note: The net contribution of all the common parts in the interior to the total line integral is 0 and only the contribution from the external contour C remains after summation

20. The maximum circulation of H per unit area as the area shrinks to zero is equal to the current density through that area

21. Two possible Amperian paths around an infinite length line of current.

22. Differential Form Integral Form Postulates of Magnetostatics in Free Space

23. Given a 3.0 mm radius solid wire centered on the z-axis with an evenly distributed 2.0 amps of current in the +a z direction, plot the magnetic field intensity H versus radial distance from the z-axis over the range 0 ≤  ≤ 9 mm. Example

24. The field from a particular line of current making up the distributed current The field from a second line of current results in the cancellation of a r component

25. This will be true for each Amperian path. AP1: So: AP2: I enc = I,