{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

solutionhw10 - Solutions to Homework Set 10 Dielectric...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Solutions to Homework Set 10 Dielectric Sphere : The solution outside will be of the form Φ out ( r, θ ) = - E 0 rP 1 (cos θ ) + C 1 r 2 P 1 (cos θ ) = - E 0 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 = 0 Φ out ∂r at r = a . These two conditions are A 1 = - E 0 + C 1 a 3 , and A 1 = - 0 E 0 - 2 0 C 1 a 3 , respectively. This pair of equations may be solved to give A 1 = - 3 0 + 2 0 E 0 C 1 = - 0 + 2 0 a 3 E 0 . Therefore E in = 3 0 + 2 0 E 0 . 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 =0 lr l P l (cos θ ) .
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}