8—Multivariable Calculus 2468.33 Find the gradient ofV, whereV = (x^2 +y^2 +z^2 )e−
√
x^2 +y^2 +z^2.8.34 A billiard ball of radiusRis suspended in space and is held rigidly in position. Very small pellets are thrown
at it and the scattering from the surface is completely elastic, with no friction. Compute the relation between the
impact parameterband the scattering angleθ. Then compute the differential scattering cross sectiondσ/dΩ.
Finally compute the total scattering cross section, the integral of this overdΩ.
8.35 Modify the preceding problem so that the incoming object is a ball of radiusR 1 and the fixed billiard ball
has radiusR 2.
8.36 Find the differential scattering cross section from a spherical drop of water, but instead of Snell’s law, use
a pre-Snell law:β=nα, without the sines. Is there a rainbow in this case?
Ans:R^2 sin 2β
/[
4 sinθ| 1 − 2 /n|]
, whereθ=π+ 2(1− 2 /n)β8.37 From the equation ( 29 ), assuming only a singlebfor a givenθ, what is the integral over alldΩofdσ/dΩ?
8.38 Solve Eq. ( 33 ) forbwhendθ/db= 0. Forn= 1. 33 what value ofθdoes this give?
8.39 If the scattering angleθ= π 2 sin(πb/R)for 0 < b < R, what is the resulting differential scattering cross
section (with graph). What is the total scattering cross section? Ans: 2 R^2
/[
π^2 sinθ√
1 −(2θ/π)^2]
8.40 Work out the signs of all the factors in Eq. ( 35 ), and determine from that whether red or blue is on the
outside of the rainbow. Ans: Look
8.41 If it suddenly starts to rain small, spherical diamonds instead of water, what happens to the rainbow?
n= 2. 4 for diamond.
8.42 What would the rainbow look like forn= 2? You’ll have to look closely at the expansions in this case. For
smallb, where does the ray hit the inside surface of the drop?
8.43 The secondary rainbow occurs because there can be two internal reflections before the light leave the drop.
What is the analog of Eqs. ( 30 ) for this case? Repeat problems 38 and 40 for this case.