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I have been trying to figure out how Gauss' law of gravitation implies that the gravitational field within a spherically symmetrical shell is zero. I have spent a very long time thinking about this and trying to find an answer online, but no luck. I would appreciate hints rather than a complete answer.

So far, some of my thoughts have been:

  • The net flux of the gravitational field of any mass not contained within some bounded surface is zero through the surface, even though the gravitational field inside the surface due to this single mass is non-zeo

  • I have thought about constructing arbitrarily small closed surfaces within the shell, but that hasn't gotten me anywhere.

In a nutshell, I realise that the difficulty I am having is that all Gauss' law tells us about is what happens at some bounding surface. It doest seem to tell us anything at all about the field within some surface. How can one extrapolate this information from Gauss' law?

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  • $\begingroup$ Because what happens at a bounding surface puts restrictions on what happens within that surface. Given a $1/r^{2}$ potential, that restriction is known as Gauss' law. $\endgroup$ – Jon Custer May 1 '17 at 20:19
  • $\begingroup$ Have you looked at "Newton's shell theorem". Google that (even on this site) and you will find a ton of information. You asked for a "hint rather than complete answer". I think that should be enough... $\endgroup$ – Floris May 1 '17 at 20:19
  • $\begingroup$ Use a closed sphere, by symmetry the field must be perpendicular to the surface and the same everywhere on the surface, so you can take the field outside the integral $\endgroup$ – user126422 May 1 '17 at 20:20
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/150238/2451 and links therein. $\endgroup$ – Qmechanic May 1 '17 at 20:26
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See if this argument works for you:

Spherical symmetry tells me that the gravitational field at any point on a spherical surface centered on the center of the sphere must be the same everywhere (pointing straight in or out, and same magnitude everywhere).

The above must be true whether you are inside the shell or outside.

The total flux through the shell must be zero (when there is no mass inside). But the flux is pointing in the same direction everywhere (relative to the surface) - so it must be zero everywhere.

Once I have determined this to be true for any point on a spherical surface, it is true for any point inside the shell - because every point at distance $r$ is part of a spherical shell at distance $r$ and must therefore have no field at its surface.

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  • $\begingroup$ Hi Floris. Thank you for your answer. I have not seen an explanation like this before and it is the first that has made sense to me. I don't know much about bumping up answers etc, but certainly think this should be in a more prominent place! $\endgroup$ – Meep May 1 '17 at 20:29

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