9—Vector Calculus 1 2799.28 In the preceding problem, what is the total energy in the gravitational field,
∫
udV? How does this (÷c^2 )
compare to the massMthat you used in setting the value ofgrasr→∞?
9.29 Verify that the solution Eq. ( 43 ) does satisfy the continuity conditions onV andV′.
9.30 Ther-derivatives in Eq. ( 38 ) can be written in a different and more convenient form. Show that they are
equivalent to
1
r
∂^2 (rV)
∂r^29.31 The gravitational potential from a point massM is−GM/rwhereris the distance to the mass. Place
a single point mass at coordinates(x,y,z) = (0, 0 ,d)and write its potentialV. Write this expression in terms
of spherical coordinates about the origin,(r,θ), and then expand it for the caser > din a power series ind/r,
putting together the like powers ofd/r. Do this through order(d/r)^3. Express the result in the language of
Eq. (4.41).
9.32 As in the preceding problem a point massM has potential−GM/rwhereris the distance to the mass.
The mass is at coordinates(x,y,z) = (0, 0 ,d). Write its potentialV in terms of spherical coordinates about the
origin,(r,θ), but this time taker < dand expand it in a power series inr/d. Do this through order(r/d)^3.
Ans:(−GM/d)[1 + (r/d)P 1 (cosθ) + (r^2 /d^2 )P 2 (cosθ) + (r^3 /d^3 )P 3 (cosθ) +···]
9.33 Theorem: Given that a vector field satisfies∇×~v= 0everywhere, then it follows that you can write~vas
the gradient of a scalar function,~v=−∇ψ. For each of the following vector fields find, probably by trail and
error, a functionψthat does this. Firstdetermine is the curl is zero, because if it isn’t then your hunt for aψ
will be futile. You’re welcome to try however — it will probably be instructive.
xyˆ^3 + 3y xyˆ^2 , xyˆ cos(xy) +ˆy xcos(xy),
xxˆ^2 y+y xyˆ^2 , ˆxy^2 sinh(2xy^2 ) + 2ˆy xysinh(2xy^2 )9.34 A hollow sphere has inner radiusa, outer radiusb, and massM, with uniform mass density in this region.
Find (and sketch) its gravitational fieldgr(r)everywhere.
(b) What happens in the limit thata→b? In this limiting case, graphgr. Usegr(r) =−dV/drand compute