6 Example For an infinite line filament with current I afa1rFor an infinite line filament with current I(a1=180o and a2=0o):
7 PE. 7.1 Find H at (0,0,5) Due to current in (figure): where a1=90o and zDue to current in (figure):where a1=90o and(0,0,5)y10A1x
8 Circular loop Defined by Apply Biot-Savart: dlRzydHzdHrxDefined byApply Biot-Savart:Only z-component of H survives due to symmetry:
9 Ampere’s Law Simpler Analogous to Gauss Law for Coulombs For symmetrical current distributions
10 Ampere’s LawWe define an Amperian path where H is constant.
11 Infinitely long coaxial cable zFour cases:1) For r<a
12 Infinitely long coaxial cable zFour cases:2) For a<r<b
13 Infinitely long coaxial cable Four cases:3) For b<r<b+cz
14 Infinitely long coaxial cable Four cases:4) For r>b+c
15 Sheet of current distribution zCross section is a Line!4baThe H field is given by:1yK [A/m]32The H field on the Amperian path is given by:x
16 PE. 7.5 Sheet of currentPlane y=1 carries a current K=50 az mA/m. Find H at (0,0,0).yK =50 mA/mz-x
17 A toroidA 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.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.Fields stay inside core, no interference.
18 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 doesn’t exist.
19 Maxwell’s Equations for Static Fields Differential formIntegral FormGauss’s Law for E field.Gauss’s Law for H field. Nonexistence of monopoleFaraday’s Law; E field is conserved.Ampere’s Law
20 Magnetic Scalar and Vector Potentials, Vm & 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:
21 The magnetic vector potential, A, is It can be shown that:The magnetic vector potential A is used in antenna theory.Substituting into equation for Magnetic Flux:This is another way of finding magnetic flux.
22 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, 0≤x≤1, -1≤y ≤4Answer :