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

Magnetic Fields

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

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

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

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

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

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

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.

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

Cross product

Fundamental Postulates of Magnetostatics in Free Space

Law of conservation of magnetic flux

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

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

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

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

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

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

Differential Form Integral Form Postulates of Magnetostatics in Free Space

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

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

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