Finding the Optimal Ratio for Two Concentric Spheres in a Capacitor

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Discussion Overview

The discussion revolves around determining the optimal ratio of the radii of two concentric spheres in a capacitor, specifically aiming to minimize the electric field between the spheres. The context includes theoretical calculations and conceptual understanding related to capacitors and electric fields.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • One participant expresses uncertainty about how to begin the problem of calculating the radius ratio for the spheres.
  • Another participant references a previous discussion related to concentric spheres and provides a link to it.
  • A participant cites a formula for capacitance involving concentric spheres, indicating a potential starting point for calculations.
  • One participant shares their approach to the problem, providing equations for capacitance and voltage, but notes a lack of progress in deriving a useful relationship.
  • Another participant mentions the relationship between electric field strength and radius, suggesting that as the radius increases, the electric field decreases.
  • A participant attempts to derive a relationship for the electric field based on charge and geometry but seeks clarification on how to express the radius ratio.
  • A later post shares a scanned solution from a professor, indicating a possible resolution or guidance but leaves the interpretation open to others.

Areas of Agreement / Disagreement

Participants express various approaches and equations related to the problem, but there is no consensus on the optimal ratio or a clear path to the solution. The discussion remains unresolved with multiple viewpoints and methods presented.

Contextual Notes

Some participants reference specific equations and concepts but do not clarify all assumptions or dependencies, leaving certain mathematical steps and definitions unresolved.

Xinthose
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Homework Statement



It's desired to build a capacitor which has two concentric spheres separated by a dielectric of high permittivity, low loss, and high dielectric strength. Calculate the ratio of sphere b's radius to sphere a's radius which produces the lowest electric field between the spheres.

Not sure how to start this one. Thank you for any help.
 
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I do know, from Wikipedia, that concerning concentric spheres, Cap = 4 (pi) ε / ( (1 /a) - (1/b) )
 


So what I've done on this problem so far is
C=[4pi(epsilon nought)][(ab)/(b-a)]
and
V=(Q/[4pi(epsilon nought)])(b-a)/(ab)
and plug it in Q=CV
i just get Q=Q

Conceptually
from E=kq/r^2 as the radius goes up the E field goes down
so the ratio from B to a would approach infinity or
A should be much less then A?
 


The electric field E = V/d where d is the distance between the plates.
 


Alright, but you eventually get E = Q / (4 * pi * ε * a * b) ; so how would you get a ratio from that from b to a ?
 


You failed me Physics Forums; here is the scanned answer from my professor's solution set given to us after the test; I hope that this will help someone else out there; Make of it what you will; his handwriting is kind of hard to read

http://i633.photobucket.com/albums/uu57/Xinthose/scan0002.jpg

http://i633.photobucket.com/albums/uu57/Xinthose/scan0003.jpg

or if you prefer to see it on the forum

page 1
scan0002.jpg

page 2
scan0003.jpg
 
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