9—Vector Calculus 1 2769.15 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 mass densityρ 0 betweenz=±d/ 2 and~g=gz(z)ˆz.
Use Eqs. ( 32 ) 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:gz= +2πGρ 0 d, (z <−d/ 2 )
9.16 Use Eqs. ( 32 ) to find the gravitational field of a very long solid cylinder of uniform mass densityρ 0 and
radiusR. (Assume it’s infinitely long.)
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. (8.16) you have an approximate expression for the gravitational field of Earth, including the effect
of the equatorial bulge. Does it satisfy Eqs. ( 32 )? (r > REarth)
9.19 Compute the divergence of the velocity function in problem 3 and integrate this divergence over the volume
of the box specified there.
9.20 The gravitational potential, equation ( 37 ), for the case that the mass density is zero says to set the Laplacian
Eq. ( 38 ) equal to zero. Assume a solution to∇^2 V = 0to be a function of the spherical coordinatesrandθ
alone and that
V(r,θ) =Ar−(`+1)f(x), where x= cosθShow that this works provided thatfsatisfies a certain differential equation and show that it is the Legendre
equation of Eq. (4.16) and section4.9.
9.21 The volume energy density,dU/dV in the electric field is 0 E^2 / 2. The electrostatic field equations are the
same as the gravitational field equations, Eq. ( 32 ).
∇.E~=ρ/ 0 , and ∇×E~= 0