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hw6_p5406_s07

# hw6_p5406_s07 - M using the LaPlace equation However now...

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HW 6 Phys5406 S07 due 3/15/07 1) The 3D G D obtained in spherical coordinates between the two radii a, b must be used if there are no other symmetries. If the boundary conditions and volume charge density are independent of φ you can use a 2D Green function. Calculate it. If the boundary conditions and volume charge density are also independent of θ you can use a 1D Green function. Calculate it. 2) Consider the case where there is no volume charge density and Φ( a, θ, φ ) = V and Φ( b, θ, φ ) = V , where V, V are constants. Solve the problem via Gauss’s law and via Green’s functions and see that you get the same potential for a < r < b . Do the same if there is uniform volume charge density ρ in this space. 3) Consider again the case of the uniformly magnetized sphere. In class we solved for Φ
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Unformatted text preview: M using the LaPlace equation. However, now that you know the expansion of 1 | r-r | in spherical coordinates you can use the following equation derived in class, Φ M = 1 4 π [ Z S da n · M ( r ) | r-r |-Z V dV 1 | r-r | ∇ · M ( r )] . (9 . 16) (1) Show that you obtain the same result using the above. 4) For the uniformly magnetized sphere ﬁnd the force on each hemisphere ( z >, < o ). Do they attract or repel? 5) Solve for the potential on the ﬁrst exam, Φ M , when a sphere of radius a and permeability μ is placed in vacuum in a uniform magnetic ﬁeld B e z . 1...
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