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hw5_p5406_s10 - Y lm however if you give it a little...

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HW 5 Phys5406 S10 due 3/4/10 1) In class the Green function for the following geometry was obtained. A sphere of radius a kept at potential V ( a, θ, φ ) is surrounded by a concentric spherical shell of inner radius b kept at potential V ( b, θ, φ ). a) If V and V’ are azimuthally symmetric find the reduced Green function g 2 ( r, r , θ, θ ). b) If V, V’ are radially symmetric, find the reduced green function g 1 ( r, r ). c) Use the reduced Green function g 1 to obtain the potential in the region a r b for the case of no volume charge density and conductors. Check your answer by solving the problem using Gauss’s law. 2) Again consider both objects to be conductors and there is the following charge dis- tribution: a ring of radius c and charge Q in the x=d plane is between the the sphere and shell. Note, there is neither radial nor azimuthal symmetry. At first you might think that this is a very difficult problem where you are required to fully probe the mathematics of the
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Unformatted text preview: Y lm , however, if you give it a little thought a simpler method will emerge. 3) In class the magnetic Feld due to a sphere of radius R with uniform magnetization was obtained via solution of the Laplace equation. Now that you know the expansion of | r-r ′ | − 1 in spherical coordinates you can obtain the Felds from a surface integral, do so. 4) When a dielectric sphere of radius R and dielectric constant ǫ/ǫ = k is placed in an initially uniform electric Feld, E = E e x , it becomes polarized. JDJ on pages 157-8 shows that the polarization is P = 3 ǫ k − 1 k +2 E . a) If the sphere is set spinning with angular velocity ω e z , that is about an axis perpendicular to E , there will be a non-zero magnetic Feld everywhere, Fnd that Feld. 1...
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