potential of spherical shell

Let (ξ,η,ζ) be a point bearing a mass m, and let (x,y,z) be a variable point. If the distance between these points is r, we can define the potential of (ξ,η,ζ) at (x,y,z) as

m- ---------------m---------------- r = ∘ -------2---------2----------2. (x − ξ) + (y − η) + (z − ζ)

The relevance of this concept appears from the fact that its partial derivatives

∂ (m ) m (x − ξ) ∂ (m ) m (y − η) ∂ (m ) m (z − ζ) --- -- = − ----3----, --- -- = − ----3----, --- -- = − ----3---- ∂x r r ∂y r r ∂z r r

are the components of the gravitational force with which the material point (ξ,η,ζ) acts on one unit mass at the point (x,y,z), provided that the measure units are chosen suitably.

The potential of a set of points (ξ,η,ζ) is the sum of the potentials of the individual points, and therefore may be expressed by an integral.

We determine the potential of all points (ξ,η,ζ) of a hollow ball, where the matter is located between two concentric spheres with radii R₀ and R, with R > R₀. The density is assumed to be a continuous function ϱ = ϱ(r) of the distance r from the center O. Let a be the distance from O to the point A at which the potential is to be determined. We choose O as the origin and the ray OA as the positive z-axis.

For obtaining the potential at A, we integrate over the spherical shell R₀ ≤ r ≤ R. We use spherical coordinates r, φ, and ψ related to Cartesian coordinates by

x = r cosφ cosψ, y = rcosφ sinψ, z = rsin φ.

The full shell is described by

R ≤ r ≤ R, − π-≤ φ ≤ π-, 0 ≤ ψ < 2π. 0 2 2

The law of cosines gives

∘ ------------------ P A = r2 − 2ar sinφ + a2.

Thus the potential is

V (a)	= ∫ _R₀^R ∫ ₋ ∫ ₀^2π dψ dφdr
	= 2π ∫ _R₀^Rϱ(r) r dr ∫ ₋.	(1)

The Jacobian factor is

| | ||∂(x,y,-z)||= r2cosφ. |∂(r,φ,ψ )|

For the remaining angular integral,

∫ ₋	= −_φ=−^φ=
	= .	(2)

Accordingly, there are two cases.

1^∘. The point A is outside the hollow ball, i.e. a > R. Then |r − a| = a − r for all r ∈ [R₀,R]. The value of the integral in (2) is 2r∕a, and (1) becomes

∫ R V (a) = 4π- ϱ (r )r2dr = M-, a R0 a

where M is the mass of the hollow ball. Thus the potential outside the hollow ball is exactly the same as if all its mass were concentrated at the center. The corresponding attractive force is represented by

V ′(a) = − M--. a2

2^∘. The point A is in the cavity of the hollow ball, i.e. a < R₀. Then |r −a| = r −a throughout the interval of integration. The value of (2) is 2, and (1) yields

∫ R V (a) = 4π ϱ(r)rdr, R0

which is independent of a. Thus the potential of the hollow ball, when the density depends only on the distance from the center, is constant inside the cavity, and the hollow ball exerts no net gravitational force on a mass inside it.

References

[1] Ernst Lindelöf, Differentiali- ja integralilasku ja sen sovellutukset II, Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1932).

Anyone with an account can edit this entry. Please help improve it!

"potential of spherical shell" is owned by pahio. [ full author list (2) ]
(view preamble)