Solving Spherical and Cylindrical Capacitors for Inner Radii

1. The problem statement, all variables and given/known data
I have two homework problems, both of which require me to solve the equation for the capacitance of a capacitor for the inner radius of the capacitor (one cylindrical, one spherical). This shouldn't be a problem, but I think my algebra is screwy.
a = inner radius
b = outer radius
C = capacitance
h = height of capacitor

2. Relevant equations
Capacitance of a Spherical Capacitor:
(i) C=4[tex]\pi[/tex][tex]\epsilon_{0}[/tex]ab / (b-a)

Capacitance of a Cylindrical Capacitor:
(ii) C=2[tex]\pi[/tex][tex]\epsilon_{0}[/tex]h / ln(b/a)

3. The attempt at a solution
My attempts at solving these for a are as follows:
(i) a = b*C / (C + 4*[tex]\pi[/tex]*[tex]\epsilon[/tex]*b)
(ii) a = b / e[tex]^{(2*\pi*\epsilon*h / C)}[/tex]
 
Last edited:

Delphi51

Homework Helper
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Both are correctly derived!
What is the h in the spherical formula?
 
The equations are solved for a correctly?

There is no h in the spherical formula.
 

Delphi51

Homework Helper
3,407
10
Capacitance of a Spherical Capacitor:
(ii) C=2[tex]\pi[/tex][tex]\epsilon_{0}[/tex]h / ln(b/a)

(ii) a = b / e[tex]^{(2*\pi*\epsilon*h / C)}[/tex]
An h is showing in each formula on my screen. Just before "/ ln(b/a)" in the first formula.
 
whoops, i had them labeled wrong. (i) is the capacitance of a spherical capacitor, and (ii) is the capacitance of a cylindrical capacitor. h is the height of the capacitor. Hm, I must be entering values into my calculator incorrectly... rechecking now.
 
Yeah, I solved the cylindrical one (ii) correctly, I just just plugging values in incorrectly.

The spherical one is still giving me issues though, I'm getting

a = b + (C / (4 * Pi * E)), which means that a will always be greater than b, an impossible situation when a is the inner radius. So I think that somewhere I've got my algebra wrong, specifically a minus sign missing somewhere.
 

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