# Solving Spherical and Cylindrical Capacitors for Inner Radii

#### meph11

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.
C = capacitance
h = height of capacitor

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

Capacitance of a Cylindrical Capacitor:
(ii) C=2$$\pi$$$$\epsilon_{0}$$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*$$\pi$$*$$\epsilon$$*b)
(ii) a = b / e$$^{(2*\pi*\epsilon*h / C)}$$

Last edited:

#### Delphi51

Homework Helper
Both are correctly derived!
What is the h in the spherical formula?

#### meph11

The equations are solved for a correctly?

There is no h in the spherical formula.

#### Delphi51

Homework Helper
Capacitance of a Spherical Capacitor:
(ii) C=2$$\pi$$$$\epsilon_{0}$$h / ln(b/a)

(ii) a = b / e$$^{(2*\pi*\epsilon*h / C)}$$
An h is showing in each formula on my screen. Just before "/ ln(b/a)" in the first formula.

#### meph11

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.

#### meph11

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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