How to calculate the capacitance?

[stefen]

Hi,everyone

The downside is a flat surface and the upside is a spheric surface,
How to calculate the capacitance between them?

 

[Externet]

Hi.
With integral calculations; the sum of capacities of infinitesimal areas projected from the sphere to the flat surface, plus taking in account the dielectric constant in the capacitance formula.
Miguel

 

[duncan94022]

One of those situations where you can spend a lot of time and effort trying to calculate it, or just build a prototype and measure it. Even if you have to iterate several timess on build/measure it will probably be faster.

If you have to calculate it, C=0.224K(A/d), where K is the dielectric constant, A is the plate area, and d is the distance between the plates. To make this simple (which is of course what will reduce the accuracy), assume that the capacitive effects are between the plate and the area directly under the sphere. Let's call the radius of the sphere r and the distance between the plate and the center of the sphere P. The simplified area of the plate is Pi*r^2 (in reality it will be larger due to fringing). The distance between the sphere and the plate is P-4*Integral(z direction)Integral(x direction)[(r^2-x^2)^0.5]dxdz. It is 4 times the integral because it is easiest to integrate from 0 to r (one quarter of the hemisphere) and multiply by 4. Since the integral of (r^2-x^2)^0.5 is (Pi/4)r^2. Integrate in the z direction and you get d=P-(Pi/4)r^3.

Final formula: C=(0.224K)(2Pi*r)/(P-(Pi*r^3)/4)

 

[melad]

if one plate of capacitor is less in are than the other , which are takes the effect in capacitance value the large or small?

Added after 1 minutes:

i one plate of capacitor is less in area

 

[tjr]

c=εA/d
whre A is the area between plates and d is the distance between plates

 

[DugalMc]

Hi, i'd do this :

give -q charge to one surface and q to the other ...

then calculate the potential difference between this two surface due the charge
(it'll be something like constant times q) then knowing Δv and q you get c = q/Δv(q) (Δv(q) = constXq) ... hope it works