H = magnetic field intensity [A/m] B = magnetic field density [Teslas] In free space the permeability is:
Magnetic Field Biot-Savart Law States that:
Example For an infinite line filament with current I ( 1 =180 o and 2 =0 o ): 2 1 a
PE. 7.1 Find H at (0,0,5) Due to current in (figure): where 1 =90 o and 10A 1 y z x 1 (0,0,5)
Circular loop Defined by Apply Biot-Savart: Only z -component of H survives due to symmetry: dl R z y dH z dH x
Amperes Law Simpler Analogous to Gauss Law for Coulombs For symmetrical current distributions
Amperes Law We define an Amperian path where H is constant.
Infinitely long coaxial cable Four cases: 1) For
Infinitely long coaxial cable Four cases: 2) For a<
Infinitely long coaxial cable Four cases: 3) For b<
Infinitely long coaxial cable Four cases: 4) For >b+c
Sheet of current distribution K [A/m] b a x z y Cross section is a Line! The H field on the Amperian path is given by: The H field is given by:
PE. 7.5 Sheet of current Plane y=1 carries a current K=50 a z mA/m. Find H at (0,0,0). K =50 mA/m -x y z
Toroidal inductors can have higher Q factors and higher inductance than similarly constructed solenoid coils. This is due largely to the smaller number of turns required when the core provides a closed magnetic path. The magnetic flux in a toroid is largely confined to the core, preventing its energy from being absorbed by nearby objects, making toroidal cores essentially self-shielding. A toroid A circular ring-shaped magnetic core of iron powder, ferrite, or other material around which wire [N- loops] is coiled to make an inductor. Toroidal coils are used in a broad range of applications, such as high-frequency coils and transformers. Fields stay inside core, no interference.
Magnetic Flux Density, B The magnetic flux is defined as: which flows through a surface S. The total flux thru a closed surface in a magnetic field is: Monopole doesnt exist.
Maxwells Equations for Static Fields Differential form Integral Form Gausss Law for E field. Gausss Law for H field. Nonexistence of monopole Faradays Law; E field is conserved. Amperes Law
Magnetic Scalar and Vector Potentials, V m & A When J=0, the curl of H is =0, then recalling the vector identity: We can define a Magnetic Scalar Potential as: The magnetic Vector Potential A is defined:
The magnetic vector potential, A, is It can be shown that: Substituting into equation for Magnetic Flux: The magnetic vector potential A is used in antenna theory. This is another way of finding magnetic flux.
P.E. 7.7 A current distribution causes a magnetic vector potential of: Find : B at (-1,2,5) Answer: Flux thru surface z=1, 0x1, -1y 4 Answer :