Talk:Shell theorem

message: I'm pretty sure I've seen intuitively simpler proofs of this, if not necessarily shorter. Though unfortunately I can't remember when and have no sources, it might even be a memory from lectures when I was a maths undergraduate over 30 years ago.
After thinking it over myself, here is an intuitive proof for the second result, inside a spherical shell. Only problem is, I have no ref. to published source so it is original research.
Anyway here goes - this is just a sketch proof which you could formalise using epsilon / delta methods: 
Take a point A anywhere inside the shell and you want to calculate the force on A. The idea is to map any point P on the shell S to the it's "opposite point" P' by drawing a line from P to P' through A. Call this mapping M.
Now you can take any plane X through A. This divides the shell S into two parts call them S1 and S2.
In this next reasoning you have small error terms throughout, and to complete the proof, you have to show that these can be made as small as you like by making the subdivisions as small as you like.
So, now take any small region r1 in S1. By the mapping M, r1 maps to a region r2 in S2. Then the surface areas, and so the masses, of r1 and r2, are proportional to the square of the distance from A (because they are related by the mapping M). Since it's an inverse square law, then the gravitational forces on A from r1 and r2 are also proportional to the inverse square of the distance. So the two forces are the same (and opposite in direction) so cancel each other out.
You can divide S1 into as many regions as you like, as small as you like, and the map M maps this dissection of S1 to a corresponding dissection of S2.So you can see that the force on A from S1 and from S2 cancel each other out.
This is true for any choice of X. So the total force on A due to the spherical shell is zero.
The nice thing about this way of proving it is that it also makes it clear why the force law has to be an inverse square for the result to hold, in a nice intuitive way. Of course this isn't a rigorous proof yet, just a sketch but wouldn't be that hard to make it rigorous. I won't bother here, as it would count as original research anyway, and someone must have found and published this proof. Anyone know a reference for it? Robert Walker (talk)
:I would like to add some thought because this reasoning is still incomplete. While it is true that the surface area is proportional to the inverse square of the distance, there is also another factor - the angle between the surface and the line (between r1 and r1r2 and between r2 and r1r2). Fortunately, because this is a sphere, the two angles are always equal, which means the surface area would be equal and the gravity would be equal. This explains why the shape must be the sphere for the theorem to work.  Natso26 (talk) 17:08, 18 November 2013 (UTC)
thread_title: Simpler proof of the result inside a spherical shell

message: In the very first part of this article, it says that the distance from a band to the test mass in s, seen here:
Am I missing something, or is this distance along the band to the test mass variable? There is not one distance between the two but rather an infinite amount of different distances. 
One may be seen at the end where the slab touches the circle's perimeter, another distance may be seen where the radius, r, and the slab intersect. If it is extended into 3 dimensions, then you get even more points were s is variable.
This approximation would only be valid if the test mass were extremely far away from the sphere.
Am I correct? If not, what am I missing?
I assume it is an approximation wrt General Relativity, but that should be clarified.
:what you are missing is that that example doesn't talk about about normal spheres, it talks about hollow spheres.62.238.198.8 (talk) 02:50, 1 June 2010 (UTC)
replies: text: The actual thing you are missing is the understanding of what d(theta) means. When solving this by normal methods d(theta) will have some length to make the integration finite and possible to do, this will lead to approximations. But thanks to another little invention by Newton (or the other guy liezben or something) called integral calculus, when we take the integral of something such as this it makes the length d(theta) go down to 0, so there will be no variation in s and will give an exact answer.Washyleopard (talk) 18:56, 31 July 2012 (UTC)
thread_title:  Is this an approximation??? 

message: Just want to make sure my statement that "the gravitational attraction force varies linearly from 0% at the inner shell boundary to 100% (for the mass of the given sphere) at the out shell boundary" is correct.  Do you guys agree ? StuRat 02:10, 11 March 2006 (UTC)
replies: text: Not linearly. It depends on the density, and even if the density is uniform, it's a cubic function, not linear. —Keenan Pepper 03:51, 11 March 2006 (UTC)
replies: text: OK, I added that to the article, which is really starting to look like something now.  It now says it's a cubic function inside the wall of a thick, uniform density spherical shell and a linear function inside a uniform density solid sphere.  Thanks, guys, for helping me expand this stub into an article. StuRat 20:17, 11 March 2006 (UTC)
replies: text: It's cubic for a solid sphere too. —Keenan Pepper 21:03, 11 March 2006 (UTC)
replies: text: Hmm, Patrick put that part in, so you two apparently disagree on that.  I think that the article should say it's either linear or cubic, so please put that back in, when you two decide which it is. StuRat 06:45, 12 March 2006 (UTC)

text: Thanks for sorting that out, and improving the article so much. —Keenan Pepper 18:37, 23 April 2006 (UTC)
thread_title:  Thick spherical shells 

message: Sin(θ)dθ = (s/rR)ds
seems to have nothing to do with the law of cosines... or I'm missing something.
could someone explain it to me?
also, is ds the rate of change of s?  
Because I get lost when it becomes part of the notation for the integration later in the page... though I suppose that the rate of change of s per s is 1...
:The law of cosines says for a triangle with sides a, b, c, and an angle \alpha opposite a, the following relationship holds:
::a^2 = b^2 + c^2 - 2bc\cos\alpha
:In this case we have a triangle with sides R, r, s, with angles \phi, \theta opposite R, s respectively.  Therefore, 
::s^2 = r^2 + R^2 - 2rR\cos\theta
:giving s as a function of \theta.  Differentiating both sides with respect to \theta:
::\frac{d}{d\theta} \left(s^2\right) = \frac{d}{d\theta} \left(r^2 + R^2 - 2rR\cos\theta\right)
::2s \frac{ds}{d\theta} = 2rR\sin\theta
::\frac{s}{rR} ds = \sin\theta \;d\theta
:ds is the infinitesimal increase in s that happens as a result of the infinitesimal increase d\theta in \theta.  s starts at a minimum when \theta = 0 and increases until it reaches a maximum at \theta = \pi (although for convenience's sake we're actually integrating with respect to s instead).  From min to max the integration looks kind of like this:
:
:s is the distance from m to each little bit of mass in the shell at a given value of \theta; the sum total of all those little bits form a circle like a latitude line on the surface of the shell, which I'm calling dM.  It may not be immediately obvious, but s is always strictly increasing along the interval \theta = [0,\pi], even when m is inside the shell.  — Xaonon (Talk) 10:32, 29 October 2006 (UTC)
thread_title:  Third part of "By the law of cosines," 

message: Maybe I'm missing something but to me the startng point of this proof is not obvious. I do not see where the first line 
:dF_r = \frac{Gm \;dM}{s^2} \cos\phi
comes from. To me it does not seem to be something easily seen from the diagram. Is it possible to state how this is derived?
Pagarus 08:45, 14 November 2006 (UTC)
replies: text: The shaded band is circularly symmetric about the line r, so the radial components of the force on m from each little bit of the band will cancel themselves out.  Only the x component will be nonzero, and that's \cos\phi of the whole force.  — Xaonon (Talk) 21:12, 15 November 2006 (UTC)
thread_title:  Starting point 

message: I believe that this article can be made much better if we give it a rewrite to increase the audience capable of comprehending it. I'm pretty sure that the high math content in the article drives off more people than it helps, so increasing the amount of plain English should help. BioTube 03:26, 5 January 2007 (UTC)
thread_title:  Less of the arcane; more of the legible 

message: Look, am I missing something here, or is this quite a simple idea that has been endlessly complicated. Is this on the right track? Imagine a hollow, heat-proof sphere at the very centre of the Earth, and a man in that sphere. Now imagine that the Earth is uniformly dense. Gravity is pulling on that man from all sides equally, therefore all gravitation attraction is cancelled out, and the man floats weightlessly in his little sphere. There can be no other logical alternative. I remember working that out when I was 10 without recourse to Newton. Yeah, sure, give us the math, but please be so good as to state the concept to be discussed initially and with a minimum of jargon and handwaving. Myles325a (talk) 03:36, 27 November 2008 (UTC)
replies: text: It also works anywhere inside the shell. You can be ten miles north of center and you'll still be floating there, not falling in any direction  —Preceding unsigned comment added by 24.162.159.61 (talk) 04:34, 4 February 2009 (UTC)

text: On the contrary, it's not obvious. If the man is in the center, then yes, there is equal mass on all sides. What if the man was several thousand miles away from the center, however? Imagine a plane P passing through that man that is parallel to a plane passing through the center. There is going to be a lot more mass on one side of P--specifically, the side that contains the center of the shell--than the other side, which would suggest that any man inside the shell would always be pulled toward the center of the shell. But contrary to intuition, he wouldn't be, and that's why this math is necessary. —Lowellian (reply) 18:55, 20 April 2010 (UTC)
thread_title:  Is it as simple as this? 

message: Not sure if it is done erroneously or blatantly but there is a difference between “force on the particle” and “force between the particles”. The net force on the point mass at exact center of the shell (empty) is zero but it will remain there forever because of balanced gravity.  Here in this case F=GMm/r^2=infinity, because the on center distance of the shell mass and the point mass coincides,i.e. r=0. 96.52.178.55 (talk) 22:06, 15 June 2009 (UTC)
thread_title:  Inside a Shell 

message: The derivation of the shell theorem as given from the version 23 April 2006 is perfect. Since then there is no improvement except possibly the animation of the figure.
The sections "Thick Shell" and "Solid Sphere" present from the first stubs are a bit ridiculous, though! If proven for one shell it applies for the infinite number of infinite thin shells below the point mass as well as the infinite this shells above the point mass (when inside the sphere!).  One comment saying this is sufficient, no new headings! 
The gravitation in a thin vertical tunnel to the center of the Earth would therefore decrease linearly relative the value at the Earth surface, the force being proportional to the distance to the middle of the Earth. The volume corresponding to shells below the point mass is proportional to d**3 where d is the distance to the Earth center.
The article implies -- even states -- that the Shell Theorem provides an exact model of gravity of a perfect sphere.  That is not true.  Newtonian gravity is an approximation even on a perfectly uniform solid sphere.  The Schwarzschild Metric is a better approximation, exact aside from rotation.  — Preceding unsigned comment added by Fairandbalanced (talk • contribs) 19:58, 16 July 2012 (UTC)
thread_title:  Acessment of a article 

message: ++++++++++++++++++++++++++++++++
==== Outside the shell ====
A solid, spherically symmetric body can be modeled as an infinite number of concentric, infinitesimally thin spherical shells. If one of these shells can be treated as a point mass, then a system of shells (i.e. the sphere) can also be treated as a point mass.  Consider one such shell:
:Note:  d\theta in the diagram refers to the small angle, not the arclength.  The arclength is Rd\theta.
Applying Newton's Universal Law of Gravitation, the sum of the forces due to mass elements in the shaded 
band is
:dF = \frac{Gm \;dM}{s^2}. 
However, since there is partial cancellation due to the vector nature of the force, the leftover component (in the direction pointing toward m) is given by
:dF_r = \frac{Gm \;dM}{s^2} \cos\phi.
The total force on m, then, is simply the sum of the force exerted by all the bands.  By shrinking the width of each band, and increasing the number of bands, the sum becomes an integral expression:
:F_r = \int (dF_r).
Since G and m are constants, they may be taken out of the integral:
:F_r = Gm \int \frac{dM \cos\phi} {s^2}. 
To evaluate this integral one must first express dM as a function of d\theta
The total surface of a spherical shell is 
:4\pi R^2 
while the surface of the thin slice between \theta and \theta + d\theta is 
:2\pi R^2\sin\theta \,d\theta.
If the mass of the shell is M one therefore has that
:dM = \textstyle\frac{1}{2} M\sin\theta \,d\theta.
and
:F_r = \frac{GMm}{2}  \int \frac{\sin\theta  \cos\phi} {s^2}\,d\theta 
By the law of cosines,
:\cos\phi = \frac{r^2 + s^2 - R^2}{2rs}
:\cos\theta = \frac{r^2 + R^2 - s^2}{2rR}.
These two relations link the 3 parameters \theta, s and \phi that 
appear in the integral together. When \theta increases from 0 to 
\pi radians \phi 
varies from the initial value 0 to a maximal value to finally return to zero for \theta = \pi.
s on the other hand increases from the initial value r - R to the final value 
r + R when \theta increases from 0 to \pi radians. To find a 
primitive function to the integrand one has to make s the
independent integration variable instead of \theta
Performing an implicit differentiation of the second of the "cosinus law" expressions above yields
:\sin\theta \;d\theta = \frac{s}{rR} ds.
and one gets that
:F_r = \frac{GMm}{2rR}  \int \frac{\cos\phi} {s}\,ds 
where the new integration variable s increases from r - R to
r + R.
Inserting the expression for \cos\phi using the first of the "cosinus law" expressions 
above one finally gets that
:F_r = \frac{GMm}{4r^2 R} \int \left( 1 + \frac{r^2 - R^2}{s^2} \right) ds.
A primitive function to the integrand is 
:  s - \frac{r^2 - R^2}{s}
and insering the bounds r - R , r + R for the integration
variable s in this primitive function one gets that
:F_r = \frac{GMm}{r^2}.
saying that the gravitational force is the same as that of a point mass in the center of the shell
with the same mass
==== Inside a shell ====
For a point inside the shell everything above applies with the exception of the limits for the integration 
interval for the integration variable s that in this case is from R - r to 
R + r as seen in the following figure
Insering these bounds in the primitive function  
:  s - \frac{r^2 - R^2}{s}
one gets that in this case
:F_r = 0
saying that the net gravitational forces acting on the point mass from the mass elements of the shell cancel out
+++++++++++++++++++
Stamcose (talk) 22:25, 19 December 2009 (UTC)
thread_title: Draft 1

message: For a point inside the shell everything above applies with the exception of the limits for the integration 
interval for the integration variable s that in this case is from R - r to 
R + r as seen in the following figure
Insering these bounds in the primitive function  
:  s - \frac{r^2 - R^2}{s}
one gets that in this case
:F_r = 0
saying that the net gravitational forces acting on the point mass from the mass elements of the shell cancel out
+++++++++++++++++++
Stamcose (talk) 22:25, 19 December 2009 (UTC)
thread_title:  Inside a shell 

message: Here is my proposed update/replacement of paragraphs 1-4, paragraph 5 to be kept. Any objection? If no objections until then, update in one week!
==== Outside the shell ====
A solid, spherically symmetric body can be modeled as an infinite number of concentric, infinitesimally thin spherical shells. If one of these shells can be treated as a point mass, then a system of shells (i.e. the sphere) can also be treated as a point mass.  Consider one such shell:
:Note:  d\theta in the diagram refers to the small angle, not the arclength.  The arclength is Rd\theta.
Applying Newton's Universal Law of Gravitation, the sum of the forces due to mass elements in the shaded 
band is
:dF = \frac{Gm \;dM}{s^2}. 
However, since there is partial cancellation due to the vector nature of the force, the leftover component (in the direction pointing toward m) is given by
:dF_r = \frac{Gm \;dM}{s^2} \cos\phi.
The total force on m, then, is simply the sum of the force exerted by all the bands.  By shrinking the width of each band, and increasing the number of bands, the sum becomes an integral expression:
:F_r = \int (dF_r).
Since G and m are constants, they may be taken out of the integral:
:F_r = Gm \int \frac{dM \cos\phi} {s^2}. 
To evaluate this integral one must first express dM as a function of d\theta
The total surface of a spherical shell is 
:4\pi R^2 
while the surface of the thin slice between \theta and \theta + d\theta is 
:2\pi R^2\sin\theta \,d\theta 
If the mass of the shell is M one therefore has that
:dM = \frac {2\pi R^2\sin\theta }{4\pi R^2}\,d\theta = \textstyle\frac{1}{2} M\sin\theta \,d\theta
and
:F_r = \frac{GMm}{2}  \int \frac{\sin\theta  \cos\phi} {s^2}\,d\theta 
By the law of cosines,
:\cos\phi = \frac{r^2 + s^2 - R^2}{2rs}
:\cos\theta = \frac{r^2 + R^2 - s^2}{2rR}.
These two relations link the 3 parameters \theta, s and \phi that 
appear in the integral together. When \theta increases from 0 to 
\pi radians \phi 
varies from the initial value 0 to a maximal value to finally return to zero for \theta = \pi.
s on the other hand increases from the initial value r - R to the final value 
r + R when \theta increases from 0 to \pi radians.
This is illustrated in the following animation
To find a 
primitive function to the integrand one has to make s the
independent integration variable instead of \theta
Performing an implicit differentiation of the second of the "cosinus law" expressions above yields
:\sin\theta \;d\theta = \frac{s}{rR} ds.
and one gets that
:F_r = \frac{GMm}{2rR}  \int \frac{\cos\phi} {s}\,ds 
where the new integration variable s increases from r - R to
r + R.
Inserting the expression for \cos\phi using the first of the "cosinus law" expressions 
above one finally gets that
:F_r = \frac{GMm}{4r^2 R} \int \left( 1 + \frac{r^2 - R^2}{s^2} \right) ds.
A primitive function to the integrand is 
:  s - \frac{r^2 - R^2}{s}
and insering the bounds r - R , r + R for the integration
variable s in this primitive function one gets that
:F_r = \frac{GMm}{r^2}.
saying that the gravitational force is the same as that of a point mass in the center of the shell with the same mass
==== Inside a shell ====
For a point inside the shell the difference is that for \theta equal
to zero
\phi takes the value \pi radians and s 
the value R-r. When then \theta increases from 0 to 
\pi radians \phi 
decreases from the initial value \pi radians to zero and s increases from the initial value R-r to the value R+r
This can all be seen in the following figure
Insering these bounds in the primitive function  
:  s - \frac{r^2 - R^2}{s}
one gets that in this case
:F_r = 0
saying that the net gravitational forces acting on the point mass from the mass elements of the shell cancel out
Stamcose (talk) 11:23, 20 December 2009 (UTC)
thread_title: Draft 2

message: For a point inside the shell the difference is that for \theta equal
to zero
\phi takes the value \pi radians and s 
the value R-r. When then \theta increases from 0 to 
\pi radians \phi 
decreases from the initial value \pi radians to zero and s increases from the initial value R-r to the value R+r
This can all be seen in the following figure
Insering these bounds in the primitive function  
:  s - \frac{r^2 - R^2}{s}
one gets that in this case
:F_r = 0
saying that the net gravitational forces acting on the point mass from the mass elements of the shell cancel out
Stamcose (talk) 11:23, 20 December 2009 (UTC)
thread_title:  Inside a shell 

message: Here is a site that might make a useful reference link:
http://www.sparknotes.com/physics/gravitation/potential/section3.rhtml
Peace
Olleicua (talk) 03:02, 29 October 2010 (UTC)
thread_title:  Possible references 

message: I am of the opinion that this theorem is valid only for concentric spheres of finite radius. If the universe is homogeneous, isotropic AND infinitely large, then every point in the universe is equivalent, and gravitational potential is the same at every point in the universe. Therefore, gravity from the mass contained inside a sphere of finite radius r must be cancelled by gravity from mass in the infinite expanse outside radius r. 
It is proven that each concentric thin shell of finite radius R outside of radius r contributes zero gravity at radius r, but extrapolating R to infinity does not yield the correct result. No matter how large R may be, gravity from outside of R must always cancel the gravity from inside R. 
I am not a mathematician. The world needs a more elegant proof than what I have offered. Does such a proof exist? If so, it should be included in the main article, and the word "finite" should be added where appropriate. 
[edited to add new heading--Onerock (talk) 21:10, 31 May 2011 (UTC)
replies: text: 'Proponents of big bang theory apply the shell theorem to "prove" its own underlying assumption—that all the mass in the universe is contained in a finite radius.' This is incorrect. If you have an ax to grind against Big Bang models, this is not the place to do it. The qualifications you want (finiteness of radius and empty space surrounding) are both implicit in the statement of the theorem, and do not need to be made more explicit.--75.83.69.196 (talk) 01:20, 12 November 2011 (UTC)
thread_title:  Theorem not valid for infinitely large sphere 

message: Just suffice it to say that would the praxis be amenable to hemispherical shell or quasi cross-section, disorderly holes in the shell and asymmetrical shell? 68.147.52.247 (talk) 03:31, 19 December 2011 (UTC)
thread_title:  Application of Shell Theorem 

message: Hi guys, I thought I remembered from physics III that this also works for electromagnetism so long as the charge is spread evenly throughout the sphere/ shell. I couldn't find any reference to anyone using this in EM though and want to know if I'm totally off base. Mostly, if you have an evenly charged spherical shell and put a point charge inside will it indeed feel no force as in the gravity case?
Also, does it have to be a point charge or could it be any shape.Washyleopard (talk) 19:04, 31 July 2012 (UTC)
thread_title:  Application To electromagnetism? 

message: The lede claims:"The derivations below focus on gravity, but the results can easily be generalized to the electrostatic force." The theorem's application to elecrostatics is NOT a "generalization". Application to (all) forces with inverse square distance relations is a generalization, application to a specific force is not. So perhaps "...easily be generalized and applied to..." or just "...the proof can easily be changed to apply to..."216.96.77.80 (talk) 18:06, 5 August 2015 (UTC)
thread_title:  Criticism of lede 

message: I am a mathematically 'informed' layperson. I have NEVER heard of "primitive function", as the term is used here, before! It is used 4 times in the article: WHY?!! Perhaps my ignorance reflects my decades old college calculus education, but I don't think so - the LINK given sends us to "antiderivative" with its only synonyms being primitive integral or indefinite integral (of a function). Seems to me that the interval(s) of integration are known (and is NOT -∞ to +∞) so what we are actually talking about is the definite integral, but the nuances of integration terminology are lost to me in the years since it mattered to my grade point average. I assume my ignorace of this term isn't atypical, but perhaps teachers started using this term 10 or 20 years ago in college (high school) calculus courses and Wikipedia has yet to 'catch up'? If so, that is a fight for the Antiderivative article, not here. Please change the term to one that is 'generally accepted'. Obscure terminology does NOTHING to advance Wikipedia's goals, despite some editor thinking it is "cool" or "better" (it OBVIOUSLY isn't as informative since is fails the first test: its NOT GENERALLY UNDERSTOOD, even to people who have (fairly extensive) exposure to the subject.216.96.77.80 (talk) 18:34, 5 August 2015 (UTC)
thread_title:  Primitive Function 

message: Lets assume that its true for body with constant density ( But I think it's NOT) :
Isaac Newton proved the shell theorem and stated that:
    A spherically symmetric body affects external objects gravitationally as though all of its mass were concentrated at a point at its center.
But, it CAN'T be true for bodies with approximately linearly increasing density toward its center, like Earth!
Short explanation: Mass of the body was extracted in front of integral.
It CAN'T be done if the density varies with the distance from the center of the body!!!  — Preceding unsigned comment added by Vaso (talk • contribs) 04:49, 31 January 2017 (UTC)
thread_title:  Shell Theorem and density gradient of body? 

message: In formal mathematics and physics textbooks, the symbols sqrt, * and ^ (for square root, multiplication and power, respectively), are not used. The use of these symbols makes the article look unprofessional and hard to read. --Roland (talk) 08:30, 30 March 2019 (UTC)
thread_title:  Mathematical Equations Too Informal 

message: Galaxies are mostly flat disc with a dense centre and there is no hard border. I suppose the dense centre is a mass point for the outlying stars and I suppose there is a similar effect as 'inside the sphere' for stars inside the galaxy. Is the inverse square law of gravitation still applicable? The proof does not wordt for what is essentialy a more two-dimentional object. Are there any theorems treating gravity in mostly flat disc?Smiley.toerist (talk) 21:08, 19 July 2019 (UTC)
thread_title:  Application to galaxies