PHYS 598 hw10 - 1 Dielectric Sphere Consider a solid...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Physics 498/MMA Handout 10 Fall 2007 Mathematical Methods in Physics I m-stone5 Prof. M. Stone 2117 ESB University of Illinois 1) Dielectric Sphere : Consider a solid dielectric sphere of radius a and permittivity . The sphere is placed in a electric field which is takes the constant value E = E 0 ˆ z a long distance from the sphere. Recall that Maxwell’s equations require that D and E k be continuous across the surface of the sphere. a) Use the expansions Φ in = X l A l r l P l (cos θ ) Φ out = X l ( B l r l + C l r - l - 1 ) P l (cos θ ) and find all non-zero coefficents A l , B l , C l . b) Show that the E field inside the sphere is uniform and of magnitude 3 0 +2 0 E 0 . c) Show that the electric field is unchanged if the dielectric is replaced by the polarization- induced surface charge density σ induced = 3 0 - 0 + 2 0 E 0 cos θ. (Some systems of units may require extra 4 π ’s in this last expression. In SI units D E = 0 E + P , and the polarization induced charge density is ρ induced = -∇ · P ) 2) Hollow Sphere : The potential on a spherical surface of radius a is Φ( θ, φ ). We want to express the potential inside the sphere as an integral over the surface in a manner analagous to the Poisson kernel in two dimensions. a) By using the generating function for Legendre polynomials, show that 1 - r 2 (1 + r 2 - 2 r cos θ ) 3 / 2 = X l =0 (2 l + 1) r l P l (cos θ ) , r < 1 b) Starting from the expansion Φ in ( r, θ, φ ) = X l =0 l X m = - l A lm r
Image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern