Biot-Savart's law and Vector Potential Apr. 18, 2007

+ Magnetic flux density (T); permeability and for free space

+ Biot-Savart's Law (Sect. 5.4): Given current density or current find or
for finite cross section conductor
for surface current (or )
for thin wire
Note: with and being position vector of the observation point and source point.
Example 4.9: Application of Biot-Savart's law

+ Magnetic flux (Sect. 5.3) (Wb)
Maxwell's equation for static field: is solenoidal and

+ Magnetic vector potentials (Sect. 5.3):
solenoidal and by our choice , called Coulomb's gauge

+ Vector Poisson's equation:

for finite cross section conductor
for surface current (or )
for thin wire
Note: with and being position vector of the observation point and source point.
Example 4.10: Find vector potential from current source

+ Applications:
Another way to find flux ; direction of the path follows the direction of on the surface enclosed by the path according to right hand rule
Vector potential provides an easier way to find , i.e. ; particularly for antenna design.
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HW #20 Due 4/24/07
1. Consider a uniform current density and a non-uniform current density . Find the following parameters for and : a) the current density at (-3,4,6), b) the rate of increase in the volume charge density at (1,-2,3), c) the current crossing a disk of radius 5mm placed on the xy-plane and centered at the origin.
2. Solve problem 5.6 (page 224) with following steps: a) Draw a diagram of a coaxial cable with an inner conductor with current flowing into the paper and an outer conductor with current flowing out of the paper. b) Identify number of regions in the diagram and find the current enclosed by a circular path in each region. c) Apply Ampere's law to find H field by dividing the current enclosed in each region with the length of the circular path. Notice that H should be a piece-wise function of radius of the circular path. d) Find B as a piece-wise function and plot it versus .
Extra-credit:
A long circular conductor of radius a carriers a current with , when placed along the z-axis. Find inside and outside the conductor. (Notice: although is up to a, the circular path with will still enclose current from .)