PDEs and Waves

penalty. 1. An example of the use of Green’s functions in 3 dimensions. Green’s functions are often applied to ¬nd a potential. In this context, the Green’s function represents the potential of a point charge (for electrical potential) or of a point mass (for gravitational potential). The following question is about a gravitational potential V (x)=V (x,y,z) at a point (x,y,z) due to a spherical mass. Along the way you will prove a classic result from graviational theory. Consider the force of gravity acting on a point mass m, located at x =(x,y,z) due to a point mass µ located at ξ =( ξ,η,ζ). According to Newton’s Law of Gravitation, this force F is given by F = ˆ’! γmµ |x ˆ’ ξ|2³ x ˆ’ ξ |x ˆ’ ξ| , where γ is the universal gravitational constant and xˆ’ξ is the displacement vector between the two masses. The force F can be written in terms of a potential v(x;ξ) wherev(x;ξ) is the potential at (x,y,z) due to the mass atξ =( ξ,η,ζ): F m = ˆ’ˆ‡xv(x;ξ) where the subscript x indicates that the partial derivatives are taken with respect to the x, y and z coordinates. Here v = ˆ’ γµ |x ˆ’ ξ| and so, by comparing V with the expression of the Green’s function solution to the inhomogeneous solution to the Laplace equation in in¬nite space (which we derived in lectures) we see that: ˆ‡x 2v =4µÎ³Î´(x ˆ’ ξ). Now let’s assume that we have a distribution of mass, density  = (ξ) throughout a domain „¦ where ξ ˆˆ „¦ (rather than a point mass at only one point x = ξ). Then the potential V (x,y,z) at (x,y,z) due to the distributed mass is given by: V (x,y,z)=ˆ’γ#„¦ (ξ) |x ˆ’ ξ| dξ. Copyright c ƒ 2015 The University of Sydney 1 (a) Assume the domain „¦ represents a spherical mass, radius R with a constant density  for 0 ‰¤ r ‰¤ R. (i) Explain why we can centre the spherical mass at the origin and take the point where we evaluate the potential to be (0,0,Z) without loss of generality. (ii) By writing the integral over „¦ in spherical polars and writing down an explicit expression in terms of r, Z and † for |x ˆ’ ξ|, show that when 0