F10HW05 - Problem 3.1 Two concentric spheres have radii a;...

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Unformatted text preview: Problem 3.1 Two concentric spheres have radii a; b ( b > a ) and each is divided into two hemispheres by the same horizontal plane. The upper hemisphere of the inner sphere and the lower hemisphere of the outer sphere are maintained at potential V . The other hemispheres are at zero potential. Determine the potential in the region a r b as a series in Legendre polynomials. Include terms at least up to l = 4. Check your solution against known results in the limiting cases b ! 1 and a ! 0. Solution to Laplace's equation with azimuthal symmetry: ( r; ) = 1 X l =0 A l r l + B l r l 1 P l (cos ) Consider: r = a A l a l + B l a l 1 2 2 l + 1 = V Z 1 P l (cos ) d (cos ) | {z } N l = V N l Consider: r = b A l b l + B l b l 1 2 2 l + 1 = V Z 1 P 1 (cos ) d (cos ) = V Z 1 P l ( cos ) d ( cos ) = V Z 1 P l ( cos ) d (cos ) = V Z 1 ( l ) l P l ( cos ) d (cos ) = V ( 1) l N l a l a ( l +1) b l b ( l +1) A l B l = 2 l + 1 2 V N l 1 ( 1) l 1 Solve for A l B l : A l B l = 2 l + 1 2 V N l a l a ( l +1) b l b ( l +1) 1 1 ( 1) l = 2 l + 1 2 V N l a l b ( l +1) b l a ( l +1) b ( l +1) a ( l +1) b l a l 1 ( 1) l = 2 l + 1 2 V N l a l b ( l +1) b l a ( l +1) b ( l +1) + ( 1) l +1 a ( l +1) b l + ( 1) l a l ( r; ) = 1 X l =0 b ( l...
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This note was uploaded on 12/08/2011 for the course PHYSICS 505 taught by Professor Liu during the Winter '08 term at University of Michigan.

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F10HW05 - Problem 3.1 Two concentric spheres have radii a;...

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