Boundary Conditions and inductance Apr. 25, 2007

+ Magnetic materials (Sect. 5.6): their magnetic property is measured by magnetization magnetic moment per unit volume.

+ gives the enhancement on when an external is applied. It can be thought of being generated by internal bound current:
;

+ External current:

+ Total current:

+ Susceptibility: , ,

+ Different magnetic materials (Sect 5.7): diamagnetic (), temp indep, paramagnetic () temp dep; ferromagnetic () temp dep

+ Boundary conditions (Sect. 5.9): tangential -- ; normal --
Find normal from medium 2 to 1; substitute equation of the boundary and match components
Example 4.11: Magnetic boundary conditions

+ Inductance (Sect. 5.10): Mutual with
Self with
Example 4.12: Self-inductance of a long solenoid
Example 4.13: Self-inductance of a two wire transmission line

+ Neumann's formula
Example 4.14: Mutual inductance between a loop and a very long wire

+ 1st method: Pick a convenient coordinates
Let the inductor carry current I. For mutual inductance, choose I flowing in a circuit with symmetry
Find with Biot-Savart' law or Ampere's law
Find and then

+ Magnetic energy (Sect 5.11): ;
magnetic energy density

+ 2nd method: Choose a convenient coordinates and I flowing in a circuit such that can be found easily
Find and then ,
Example 4.15: Internal inductance of a very long wire
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HW #22 (no late homework) Due 4/30/07
1. Three infinite long (thin) wires locating at (x=0, y=0), locating at (x=0, y=3) and locating at (x=4, y=3) carry currents -50A, 100A and 300A respectively along . Find the force per unit length on a) due to (also state if the force is repulsive or attractive), b) due to (also state if the force is repulsive or attractive), c) due to and . (Note: The direction of magnetic forces can be visualized by drawing a diagram. From the diagram, you need to resolve the force direction in terms of x and y components.)

2. Consider the toroidal coil (height h, inner radius a and outer radius b) with N turns in Fig. 5-15 (page 204), a) assuming air core, find , , ; b) assuming the core is filled with magnetic material with permeability in the region and , find , , ; c) Find (or ) and (or ) for b).

Extra-credit
A rectangular loop carrying current is placed parallel to an infinitely long (thin) wire carrying current as shown in Fig. 5-18 (page 209). Consider the flowing counter-clockwise along the loop. Show that force experienced by the loop is given by N.