9—Vector Calculus 1 2369.13 Mimic the calculation in section9.6for the divergence in cylindrical coordinates, computing the
curl in cylindrical coordinates,∇×~v. Ans: Eq. (9.32).
9.14 Another way to get to Eq. (9.39) is to work with Eq. (9.38) directly and to write the function
ρ(r)explicitly as two cases:r < Randr > R. Multiply Eq. (9.38) byr^2 and integrate it from zero
tor, being careful to handle the integral differently when the upper limit is< Rand when it is> R.
r^2 gr(r) =− 4 πG
∫r0dr′r′^2 ρ(r′)
Note: This is not simply reproducing that calculation that I’ve already done. This is doing it a different
way.
9.15 If you have a very large (assume it’s infinite) slab of mass of thicknessdthe gravitational field
will be perpendicular to its plane. To be specific, say that there is a uniform mass densityρ 0 between
z=±d/ 2 and that~g=gz(z)ˆz. Use Eqs. (9.36) to findgz(z).
Be precise in your reasoning when you evaluate any constants. (What happens when you rotate the
system about thex-axis?) Does your graph of the result make sense?
Ans: in part,gz= +2πGρ 0 d, (z <−d/ 2 )
9.16 Use Eqs. (9.36) to find the gravitational field of a very long solid cylinder of uniform mass density
ρ 0 and radiusR. (Assume it’s infinitely long.) Start from the assumption that in cylindrical coordinates
the field is~g=gr(r,φ,z)ˆr, and apply both equations.
Ans: in partgr=− 2 πGρ 0 r,(0< r < R)
9.17 The gravitational field in a spherical regionr < Ris stated to be~g(r) =−ˆrC/r, whereCis a
constant. What mass density does this imply?
If there is no mass forr > R, what is~gthere?
9.18 In Eq. (8.23) you have an approximate expression for the gravitational field of Earth, including
the effect of the equatorial bulge. Does it satisfy Eqs. (9.36)? (r > REarth)
9.19 Compute the divergence of the velocity function in problem9.3and integrate this divergence over
the volume of the box specified there. Ans:(d−c)av 0
9.20 The gravitational potential, equation (9.42), for the case that the mass density is zero says to
set the Laplacian Eq. (9.43) equal to zero. Assume a solution to∇^2 V = 0to be a function of the
spherical coordinatesrandθalone and that
V(r,θ) =Ar−(`+1)f(x), where x= cosθ
Show that this works provided thatf satisfies a certain differential equation and show that it is the
Legendre equation of Eq. (4.18) and section4.11.
9.21The volume energy density,dU/dVin the electric field is 0 E^2 / 2. The electrostatic field equations
are the same as the gravitational field equations, Eq. (9.36).
∇.E~=ρ/ 0 , and ∇×E~= 0
A uniformly charged ball of radiusRhas charge densityρ 0 forr < R, Q= 4πρ 0 R^3 / 3.
(a) What is the electric field everywhere due to this charge distribution?