emhw04 - R carrying charge Q , immersed in a uniform...

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Electromagnetic Theory I Problem Set 4 Due: 19 September 2011 13. We have derived the spherical harmonic Y ll ( θ, ϕ ), Y ll ( θ, ϕ ) = ( ) l l ! 2 π r (2 l + 1)! 2 2 l +1 sin l θe ilϕ , (1) normalized so that i d Ω | Y lm | 2 = 1, as a solution of the diFerential equation L + Y ll = 0 where L ± = L x ± iL y are the angular momentum ladder operators. Recall that L ± raise or lower the m value of a spherical harmonic Y lm by one unit: L ± Y lm = R l ( l + 1) m ( m ± 1) Y lm ± 1 = R ( l m )( l ± m + 1) Y lm ± 1 . (2) Using these facts, apply L - repeatedly to Y ll to derive all the Y lm for 0 m l for the cases a) l = 1, b) l = 2, and c) l = 3. You may use without proof the spherical coordinate form of the L ± , L z : L ± = ± e ± p ∂θ ± i cot θ ∂ϕ P , L z = 1 i ∂ϕ (3) 14. One can sometimes use part of the solution of a simple potential problem as the solution of an apparently more complicated problem. ±or example, any equipotential surface of the simple problem can be replaced by a conductor of coincident shape. a) ±irst, ²nd the potential everywhere outside an isolated conducting sphere of radius
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Unformatted text preview: R carrying charge Q , immersed in a uniform external electric ²eld. (Hint: Take the z-axis parallel to the external ²eld, expressing its potential in spherical coordinates, taking the center of the sphere at the origin.) b) Now consider a hemispherical conductor of radius R attached to a grounded plane (the xy-plane) (i.e. the hemisphere-plane combination are all at zero potential). The top of the hemisphere is at z = R and its center is at the origin of coordinates. The electric ²eld, in the (empty) region above this conducting surface, approaches a uniform ²eld E ˆ z far from the hemisphere. Calculate the surface charge density everywhere on the conducting surface (plane and hemisphere). c) Calculate the total charge, on the hemisphere of part b), in terms of E and R . 15. J, Problem 3.4. 16. J, Problem 3.14. 1...
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This note was uploaded on 09/25/2011 for the course PHY 6346 taught by Professor Staff during the Spring '08 term at University of Florida.

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