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F07_Practice_Final

# F07_Practice_Final - Physics 505 Practice Final Fall 2007...

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Physics 505 Fall 2007 Practice Final The final will be a three hour open book, open notes exam. Do all four problems. 1. Two point charges q and - q are located on the z axis at z = + d/ 2 and z = - d/ 2, respectively. a ) If the charges are isolated in space, show that the potential admits a Legendre expansion Φ( r, θ ) = 2 q 4 π 0 l odd r l < r l +1 > P l (cos θ ) where r < = min( r, d/ 2) and r > = max( r, d/ 2). b ) Now consider the charges to be contained inside a linear dielectric sphere of permittivity and radius a (where a > d/ 2). - q q Find the electric potential everywhere as an expansion in Legendre polynomials. 2. A point magnetic dipole m is located in vacuum pointing away from and a distance d away from a semi-infinite slab of material with relative permeability μ r . a ) Find the magnetic induction everywhere. b ) What is the force on the dipole (magnitude and direction)? 3. An infinitely long solenoid of radius a has N tightly wound turns per unit length. For a constant current I , elementary considerations tells us that the magnetic induction is uniform inside the solenoid. In cylindrical coordinates,

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F07_Practice_Final - Physics 505 Practice Final Fall 2007...

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