# Hollow spherical shell of mass or charge

Discussion in 'Physics' started by someonesdad, Apr 23, 2011.

Jul 7, 2009
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The other day I read a short, simple proof of the fact that there's no force on a mass inside a uniform shell of matter (and a similar statement applies in electrostatics). This problem is usually done in an elementary class by setting up an integral. However, this simple proof is accessible to a high school student, as it just uses simple geometry and a little trig. I wrote up a blurb on it, as it also has some interesting electrical history.

It also turns out it was a problem in my basic physics text (Resnick & Halliday) when I was in school, had I bothered to read it.

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2. ### steveb Senior Member

Jul 3, 2008
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Nice work! There is real value in understanding the force cancelation in terms of small pieces, rather than over slices of shells, full shells, or through Gauss' theorem.

3. ### steveb Senior Member

Jul 3, 2008
2,433
469

I hope people will take some time to appreciate what you've done here. I've seen many different ways to derive the nullification of gravity and electrostatic force when inside a spherically symmetric mass or charge distribution, - all based on calculus.

It's said that Newton took a long time to prove this fact, although he intuitively knew it to be true. Years later Gauss proved his useful theorem which lets us solve such problems instantaneously.

However, the very elegant viewpoint you've shown here cuts out all the abstraction and gets to the core (pardon the pun) of what is going on. Your method even simplifies the math for anyone who wants to employ calculus to the problem. You can set up the integral over half of the 4pi solid angle by pairing up the contributing portions of each direction. Rather than having to do out the integral and showing that the total integration is zero as typically happens in most methods, you are able to show that the integrand is zero right from the start. Hence, the integral is (trivially) zero by inspection. Very nice!

Jul 7, 2009
1,585
141
Thanks, steveb. I certainly can't claim the proof as my own, as I just read about it in an older book I had laying around -- and I was impressed with the simplicity and shortness of the proof. The book (by Feather, referenced in the paper) refers to an earlier volume by the same author that gives the same proof that Newton gave. I'd like to see that one too, but I don't have the book.

Like you intimate, the answer for an experienced person is to use Gauss' Law. But that also requires a bit of calculus.

What really struck me (besides the elegance of the proof) was the brilliant and elegant experiment of Cavendish. I remember reading about this in my basic physics class (or the teacher explaining it), but the elegance of what the experiment did never really registered on me. My copy of Halliday and Resnick discusses the Plimpton-Lawton experiment, done in the 1930's, which was just a more sensitive version of Cavendish's experiment done 150 years before. Their results (like Cavendish's) established that the exponent in Coulomb's Law is 2 to within nearly 1e-9. Cavendish was a recluse and only verbally communicated his experimental results to his friends at the Royal Society. His papers were only published in the year Maxwell died about 100 years later.

Such high precision experiments are the experimental foundation for Coulomb's and Gauss' Laws -- and that electrical charge on an insulated conductor resides on the outer surface, a practical result used in the Faraday cage.

The Halliday and Resnick book gives this paragraph from Benjamin Franklin, written in 1755 in a letter to a friend [sic]:
I electrified a silver pint cann, on an electric stand, and then lowered into it a cork-ball, of about an inch diameter, hanging by a silk string, till the cork touched the bottom of the cann. The cork was not attracted to the inside of the cann as it would have been to the outside, and though it touched the bottom, yet when drawn out, it was not found to be electrified by that touch, as it would have been by touching the outside. The fact is singular. You require the reason; I do not know it...
Note that Cavendish's experiment was done more than 50 years before Gauss' Law was formulated by Gauss and a decade or more before Coulomb's important experiments. Thus, a student studying physics and Maxwell's equations should at least tip his or her hat towards a brilliant experiment that lies at the foundations of one of Maxwell's equations.

By the way, some people call the book Resnick and Halliday and some call it Halliday and Resnick. My copy is in two volumes and one volume has the pair of names one way and the other volume switches them around. So if you felt like "correcting" me (like one friend did in an email), realize it can be either way.

And how come I can remember buying that book in the college bookstore back in the 60's for \$10 exactly, but I can't remember what I had for lunch yesterday?