Why do the signs alternate when using Gauss' law for gravity?

[TimeRip496]

"Gauss's theorem can be established as follows. consider an attracting particle of mass m at the point P, and let a cone of small solid angle ω be generated by radii through P. This cone cuts the surface at the points Q₁, Q₂, ... taken in order from P; the parts of the surface cut off by the cone at these points are S₁, S₂, ... and the outward-drawn normals are denoted n₁, n₂, ...

Now if P lies outside Σ (region of masses, each at points Q₁, Q₂, ...), the cone will cut S an even number of times, and the signs will be plus and minus alternately, so that the total normal force across S₁, S₂, ... will be zero. On the other hand, if P lies inside Σ, the cone will cut S an odd number of times, the sign being minus and plus alternately, so that the total normal forces across S₁, S₂, ... will be -mω."

Source: Spherical harmonic T.M.Macrobert

Help with this text! I don't understand the second paragraph as to why the cone will cut S an odd number of times, the sign being minus and plus alternately when P lies inside Σ vs when P lies outside Σ. I know the cone is just a mathematical object but I can't follow the author thought process here. How did he use the cone to cut the surface at Q₁, Q₂, ... ? Why will the normal force across S due to particle at P be zero when P is outside the mass but not inside the mass?

 

[stevendaryl]

Above is a picture illustrating the point. You have a point mass on the left, and a surface drawn on the right. The flux through a section of a closed surface is positive if the vector from the mass to the section of the surface points outward. The flux is negative if it points inward.

If the point mass is outside the surface, the positive flux through one section is canceled by the negative flux through another section. So the total flux adds up to zero. If the point mass is inside the surface, then you just have outward flux (positive).

You can make the surface more complicated by putting in "folds" in the surface. But each fold will have cancellation of the positive and negative fluxes, so folds don't change the total flux.

So the even/odd business is:

If the point mass is inside the surface, you have, for each cone coming away from the mass, one section with outward flux, plus 2 sections (one outward and one inward) for each fold. So the total number of sections intersecting the cone is odd.

If the point mass is outside the surface, you have two sections (one with positive flux and one with negative flux) plus 2 more for each fold. So the total number is even.

 

[TimeRip496]

View attachment 231296

Above is a picture illustrating the point. You have a point mass on the left, and a surface drawn on the right. The flux through a section of a closed surface is positive if the vector from the mass to the section of the surface points outward. The flux is negative if it points inward.

If the point mass is outside the surface, the positive flux through one section is canceled by the negative flux through another section. So the total flux adds up to zero. If the point mass is inside the surface, then you just have outward flux (positive).

You can make the surface more complicated by putting in "folds" in the surface. But each fold will have cancellation of the positive and negative fluxes, so folds don't change the total flux.

So the even/odd business is:

If the point mass is inside the surface, you have, for each cone coming away from the mass, one section with outward flux, plus 2 sections (one outward and one inward) for each fold. So the total number of sections intersecting the cone is odd.

If the point mass is outside the surface, you have two sections (one with positive flux and one with negative flux) plus 2 more for each fold. So the total number is even.

I am sorry I don't quite get the direction of the flux part. For the "flux through a section of a closed surface is positive if the vector from the mass to the section of the surface points outward. The flux is negative if it points inward.", how are the direction for the flux determined? How do I know whether the vector is pointing outwards vs inwards? If we are looking at gravity, should't the direction of flux be just one as gravity only attracts and flux is just the amount of field lines passing through that surface?

 

[stevendaryl]

I am sorry I don't quite get the direction of the flux part. For the "flux through a section of a closed surface is positive if the vector from the mass to the section of the surface points outward. The flux is negative if it points inward.", how are the direction for the flux determined? How do I know whether the vector is pointing outwards vs inwards? If we are looking at gravity, should't the direction of flux be just one as gravity only attracts and flux is just the amount of field lines passing through that surface?

A closed surface such as a balloon has an inside and an outside. The line from the point mass to a point on the surface has a direction. If the line is pointing inward, toward the inside of the surface, then the flux through that point on the surface is negative. If it is pointing outward, toward the outside of the surface, then the flux is positive.

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[stevendaryl]

In the figure below, I show an oddly-shaped surface. The line from the mass to the surface pierces the surface 6 times. 3 times the direction is inward--from outside the surface toward inside the surface. 3 times the direction is outward--from inside toward outside.