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AlchemistK
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Homework Statement
OK first, I didn't know where to post this since it is and is not homework,anyways, posting here would be safest.
Now, the attached scan is a proof from my (not so trustworthy) textbook and I have some doubts in it.
The first doubt is in statement S1 : Every element of charged conductor experiences a normal outwards force.
This is sort of a general doubt, why normal? That should be a special case when the body is symmetrical, right? Why is it necessary for all the electric field vectors to cancel out in way to produce a normal resultant?
Something vague to back me up : One known fact about electric field lines in conductors is that they are always perpendicular to the surface; for a curved object to have perpendicular lines, the lines themselves should be curved. Thus the lines passing through dS (in figure) must be curved and won't give a normal resultant unless the magnitude of the fields from opposite directions is equal.
Now, all of this I thought was based on one assumption I unknowingly made, that the charge density is uniform. Having non uniform charge density will change everything. So in the end the question is whether S1 is always true. Is it?
The second doubt is S2. I have no problem with them taking the field like that, just that when the proof is done, will we be able to use this for conductors of any magnitude and dimensions and use it to calculate force over a finite area rather than an area element?
Now, relating to this is a question which I think is meant to be solved using this result. (Hardworking people could try double integration but the question was ideally meant to be solved in 3 minutes) An alternative method will be Highly appreciated.
Q: A conducting spherical shell of radius R is given a charge Q. Find the force exerted by one half on the other half.
The attempt is done below but my question is once again regarding S2, we derived the result using the electric field as σ2/2ε, can it be used here too? Conversely, does this imply that the hemisphere has a field of σ2/2ε at the point where the charge of the other hemisphere can be thought of to be concentrated?
2. The attempt at a solution
df = σ2/2ε * dS
Taking the integral of dS as the projected area of the hemisphere, the center circular plane
pi*R2 (R being radius of the the hemisphere)
σ= Q/(2*2*pi*R2)
Thus, f = Q2/(32ε*pi*R2)