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Unformatted text preview: Solutions to Homework Set 10 Dielectric Sphere : The solution outside will be of the form Φ out ( r, θ ) =- E rP 1 (cos θ ) + C 1 r 2 P 1 (cos θ ) =- E r + C r 2 cos θ. while, inside Φ in ( r, θ ) = A 1 rP 1 (cos θ ) . These terms have been selected to match the asymptotically uniform electric field at infinity, to be finite at the centre of the sphere, and to all have the same angular dependence. The continuity of the tangential electric field will be ensured if Φ is continuous at the surface of the sphere. The continuity of the radial component of D requires ∂ Φ in ∂r = ∂ Φ out ∂r at r = a . These two conditions are A 1 =- E + C 1 a 3 , and A 1 =- E- 2 C 1 a 3 , respectively. This pair of equations may be solved to give A 1 =- 3 + 2 E C 1 =- + 2 a 3 E . Therefore E in = 3 + 2 E . 1 Hollow Sphere : The generating function for Legendre polynomials is 1 (1 + r 2- 2 r cos θ ) 1 / 2 = ∞ X l =0 r l P l (cos θ ) , r < 1 . If we differentiate this relation with respect to r , and then multiply the result by r , we have- r 2 + r cos θ (1 + r 2- 2 r cos θ ) 3 / 2 = ∞ X l...
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This homework help was uploaded on 01/29/2008 for the course PHYS 598 taught by Professor Stone during the Fall '07 term at University of Illinois at Urbana–Champaign.
- Fall '07