Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Physics LibreTexts

5.4.8: Hollow Spherical Shell

( \newcommand{\kernel}{\mathrm{null}\,}\)

We imagine a hollow spherical shell of radius a, surface density σ, and a point P at a distance r from the centre of the sphere. Consider an elemental zone of thickness δx. The mass of this element is 2πaσ δx. (In case you doubt this, or you didn’t know, “the area of a zone on the surface of a sphere is equal to the corresponding area projected on to the circumscribing cylinder”.)

Figure 5.9.png
FIGURE V.9

The field due to this zone, in the direction PO is

2πaσGcosθδxξ2.

Let’s express this all in terms of a single variable, ξ. We are going to have to express x and θ in terms of ξ.

We have

a2=r2+ξ22rξcosθ=r2+ξ22rx,

from which

cosθ=r2a2+ξ22rξδx=ξδξr.

Therefore the field at P due to the zone is πaGσr2(1+r2a2ξ2)δξ.

If P is an external point, in order to find the field due to the entire spherical shell, we integrate from ξ=ra to r+a. This results in

g=GMr2.

However, if P is an internal point, in order to find the field due to the entire spherical shell, we integrate from ξ=ar to a+r, which results in g=0.

Thus we have the important result that the field at an external point due to a hollow spherical shell is exactly the same as if all the mass were concentrated at a point at the centre of the sphere, whereas the field inside the sphere is zero.

Caution. The field inside the sphere is zero only if there are no other masses present. The hollow sphere will not shield you from the gravitational field of any other masses that might be present. Thus in figure V.10, the field at P is the sum of the field due to the hollow sphere (which is indeed zero) and the field of the mass M, which is not zero. Anti-grav is a useful device in science fiction, but does not occur in science fact.

Figure 5.10.png
FIGURE V.10


This page titled 5.4.8: Hollow Spherical Shell is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?