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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 JyUcr) 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
andshow 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
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 Fall '11
 drake
 Physics

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