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solutions_final

# solutions_final - Physics 604 Final Exam Fall ‘11 Dr...

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Unformatted text preview: Physics 604 Final Exam Fall ‘11 Dr. Drake 1. (40 pts ) (a) Evaluate the following integral 0° l. peg} IZIWdZ—Z—zU—COSZ) (b) The function f is deﬁned by :18 . with a branch out between —1 and 1. Give a plausibility argument why the branch cut does not need to be extended to inﬁnity. Evaluate ﬂirt) where a is real and positive. 2. {40 pts) The diﬁerential equation for the modified Bessel function is “.2 H by +zy’—(zg+u2)y:{} 4,... (a) What is the lowest order behavior of the solutions of this equation in the [\$ vicinity of z = 0? Consider all possible values of real V. (b) An integral representation of the modiﬁed Bessel function of the ﬁrst kind, 1,,(2) is Mg) 1. i Biz/2)(s+1/s) “35 _ 2m 0 31+“ where a cut extends from zero to negative infinity in the 3 plane and the contour C' wraps around the out, starting at negative infinity below the cut and ending at negative infinity above the cut. Evaluate the integral approximately for 2 large and positive. 3. (60 pts) Consider a two—dimensional cylindrical cavity of inner radius a and outer radius b that has an angular width \$0: rI’he electric potential V is mainw tained at —V0 and V5 at (,5 :— —q§0/2 and if) : gag/2, respectively, and zero at r 2 a, b. The potential V9", 923) satisﬁes Poisson’s equation .18 a is2 2 A——— —— —-—— Z V V_ T3TT8TV+ T2 8¢2V 0 (a) It is convenient to change to a new variable 8 : \$710") before attempting to solve this problem. What is the equation satisﬁed by V when expressed If in terms of 5 and Q5? (1)) Write the solution for V in terms of a separable set of eigenfunctions (Dmﬁt) and Rm(s) as follows: V(S: Cb) : Z Cmles)(Ilm(¢) _-~— Write equations for ‘1)le5) and R1145) and solve these equations using 2> boundary conditions appropriate for the solution V. Normalize the basis functions so they have unit norm. Hint: Use the symmetry in 95 to simplify the solution. 2 0 (c) Solve for cm by matching the form of the potential at the side boundaries. 4. (60 points) A liquid is heated in a hollow sphere of radius 1) by a pulse of heat at t : O at radius r r: n). The equation satisﬁed by the liquid is T . 595? M anT =- A5{t)6(r — m} where 1 8 a 2 = *w 2_ V T2 Br?” 8?‘ The temperature of the liquid is zero for t < 0 and remains zero at the boundary for all time. .—--{a) Construct a set of basis functions @5710") to describe the liquid in the cavity. :2 D W'rite the basis functions as a linearcombination of the two solutions of the Bessel equation JyU-cr) and YVUCT). Give expressions for the eigenvalues of your basis functions (in terms of the known properties of JAM) and l’AkrD and deﬁne the normalization so that the @710“) have unity norm. State why the basis functions are orthogonal {you don’t have to prove orthogonality). What is the behavior of the eigenfunctions near r = 0? Sketch the lowest three eigenfunctions. I Hint: Bessel’s equation is r2y” + ry’ + (1627"? —— 1/2)y = 0. Let qbn = gym/rU2 and-show that gn satisfies. Bessel’s equation. What is 1/? [1 g (b) Write T(7",t) as 00 T022?) 1' E Curiae) 71:1 and derive an equation for .1105). ‘W’hat is the characteristic damping rate of the eigenfunctions? [ g (c) Solve for Cn{t) and then write a solution for the complete space/time dependence of T.’ g (d) What is the lowest order non—trivial form'of the solution at late time? Ehﬂ/ EXﬂM/t g9/Zi 7Li0ﬁ§ péyS‘TS 5536/ @ 6L) EUR/615L716 ”SEQ-\$0 ~-/‘aga:j Maﬁa “fﬁzﬂz 7%Cmc If We Slﬂ§'%Z;2“{7z/ ﬂ% 9:3 “:> m [:06 7/?2€ [9.51%GFLW [177(65r‘fc/ ﬁftyfiy 111nm” <3; a go {my} gp/JL /‘[email protected]—/ Zr“? 7L0 ejyomv (577([31 / A‘sﬁyya’ar) {072\$ l:— gaffer {z MS?) C/ *3 ..... ng’é/L - J— iii-1'9: Q 2 C‘ 5&2- u \ H -’—\ 1‘1 C/US‘E I. {57¢ m to: LIW3 “:3 {KW} [ynfmzécaﬁcq {Ewan ggq? #[twc’f/‘6 ”L: ~—L --=} (‘3 I] 3:0 ﬁméé Swat” 1/29 cm 5/52;er fﬁ/C‘g [/27le I; {m MEL/3 22> 14?)“ [ﬁq%ﬂféw {0‘4 744’967 gkmct- ("um/i “:5 {Tow/am E éf/MW? ﬁ.‘ .-"H 1-13—Jgt: ﬂag)!“ 1327' 8/18 (/ng I; a»: Lh‘P ‘2) f7?) 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