Midterm 210a Winter 2004

Physics 210A - Mid-term -Winter 2004 - Prof. J. Incandela Turn work in to Dr. Chris Hill, Broida 5122 by noon on Friday, Feb. 13. Use only the following: (1) J. D. Jackson text (2) material from this course's web pages. Do not consult anyone. There's no need to derive any result found in the text, lectures, or homework problems. Merely indicate the source and proceed. Work through to the final answers.

1.) A major storm with lightning rolls in while a physicist friend of yours is out rock climbing on an overhanging cliff in France near to the CERN particle accelerator laboratory (see figure and caption below). He was planning to climb to the top of the cliff (position A) and he's pretty sure he can make it there before it starts raining. An alternative would be to return to the base (position B) and climb the cliff later but he would then have to wait until the rock dries. Assuming that the earth is at ground and the stormy sky above is at some elevated potential, estimate the ratio of the field strengths as a function of the distance from the corner positions A and B. What should he do and why? Figure not to scale. Cliff height is much greater than height of climber. Storm clouds are at a height much greater than the height of the cliff. The cliff itself can be assumed to extend a very long way. (i.e. can treat problem as 2 dimensional)

2.) a) Derive the Green function (i.e. the electrostatic potential for a charge 4πЄo) in the region between concentric, hollow, closed right circular cylinders of inside radii a and b (with a < b) and inside height L. Assume Dirchlet boundary conditions. b) What is the potential for the case where there is a uniformly charged wire with total charge Q and length L installed parallel to the cylinder axis (the wire is insulated from the end caps) at a radial position a < ϖ < b ? Closed, concentric cylinders, with all surfaces at ground.

3.) A group of astronauts encounter a peculiar planet of radius a (shown in the figure below). Two "ice" caps extend symmetrically to an angle α from the spin axis of the planet. They are made up of an unknown substance that the astronauts determine by a series of measurements to be at a uniform elevated potential V relative to the rest of the planet (which is taken to be at Ground). Surrounding the equator at a distance R > a from the planet's center is a thin ring which the astronauts find to be uniformly charged with an estimated total charge Q. Find the electrostatic potential everywhere outside the planet's surface.

Some General Relations: Pj+1(æ 1) = Pjà1(æ 1) Pj(à x) = (à 1)jP(x) dP j dx

ã â j = (x2 à1) xPj à Pjà1

1 Γ(n + 2)

=

l+m 2

2m ù √ ù cos[ 2 (l √ (2nà1)!! ù n 2

P l m(0) = (à 1)

+ m)]

Γ(l+m+1 2 ) Γ(làm+2 2 )