What is the Limiting Value of Integral in Newton's Shell Theorem for r=R?

[parshyaa]

Homework Statement

While deriving newton shell theorem for a hollow sphere of mass M , radius R and a point particle of mass m at a distance r, i am getting wrong answer while taking r=R (ie particle on the surface of spherical shell) Please tell me where i am going wrong.

Relevant Equations

F = $\ (GmM/4r^2R)\int_{r-R}^{r+R}((r'^2+r^2-R^2)/r'^2)dr'$

First i tried proving Newton shell theorem directly for r=R and solved the integral as above but still got the wrong solution.

Here i tried using general case:
Here r' is the distance of a small ring from the point particle of mass m
So my doubt is when we take r=R and then evaluate this equation, limit goes from 0 to 2R and integral gives the value (GmM/2r^2)
Which is wrong, so where am i going wrong?

 

[hutchphd]

I believe the "relevant equation" is dimensionally incorrect ?

 

[parshyaa]

I believe the "relevant equation" is dimensionally incorrect ?

Yep i had written r'instead of r'^2 and dr instead of dr', now its been edited

 

[Gordianus]

The result is O.K. The "paradox" arises from considering an infinitely thin layer of mass. Inside the shell (r<R) the field is zero; just outside the the shell is GM/R^2. There is "jump" from zero to a finite value. On the surface, the result is the average of the values inside and outside.

 

[parshyaa]

The result is O.K. The "paradox" arises from considering an infinitely thin layer of mass. Inside the shell (r<R) the field is zero; just outside the the shell is GM/R^2. There is "jump" from zero to a finite value. On the surface, the result is the average of the values inside and outside.

Wow insight is good and satisfying, is their a mathematical theory behind this?, Some kind of theorem regarding integration ambiguity of this kind?

 

[PeroK]

Wow insight is good and satisfying, is their a mathematical theory behind this?, Some kind of theorem regarding integration ambiguity of this kind?

You have a function that is discontinuous at $r = R$. Depending on the way you take the limit, you could end up with:
$$\lim_{r \rightarrow R^+} f(r), \ \ \lim_{r \rightarrow R^-} f(r)$$
Or something else. In your case it was something else, because you had an integral straddling the discontinuity, hence you got half of the right-hand limit and half of the left-hand limit.

The moral is to be careful taking a limit if the function is discontinuous. This idea will come up again in electromagnetism when looking at surface charges and surface currents.

One way round this problem (and an easier way to prove the shell theorem) is to use the potential, which is continuous at the shell boundary.

 

[vanhees71]

Wow insight is good and satisfying, is their a mathematical theory behind this?, Some kind of theorem regarding integration ambiguity of this kind?

Then you have some subtleties in vector calculus. You find the corresponding mathematical techniques in textbooks of electrodynamics, where one discusses the conditions for the fields at singular boundaries like surfaces. Then the differential operations like the divergence of a vector field have to be handled with some care, leading to the idea of surface divergences etc.

 

[parshyaa]

@PeroK ,@vanhees71, it seems this is the case of jump discontinuity

And still our solution manages to give average value, how cool is this?

 

[Gordianus]

I suggest reading a similar topic in Feynman's Lectures. Somewhere, he computed the force between the plates of a parallel plate capacitor. In this case, again, the field changes from zero (inside the plate) to a finite value (between the plates). Feynman used the average value and obtained the right value of the force. Perhaps he didn't follow the purest way, but he knew what he was doing.