Isotropic Capacitance

Richard Hull <hullr-at-whitlock-dot-com> Writes: 

> Basically, Space is a dielectric, in that it has a quality called 
> permittivity (8.8 picofarads/meter)  It is thru this good office and the 
> rest of the matter in the universe that a single terminal in outer space 
> has capacitance.  The equations for this value are all screwed up as are 

Greetings, Richard! 

  I'm a faithful reader, but very sporadic contributor to this list (mostly 
because there's rarely enough time to quickly read through everything; I'd 
contribute more if I could but find the time). In any case, I think my last 
contributions were before you joined here, so I thought I'd put in the little  
intro. Also, we met at Ed Wingate's Teslathon last summer, but who's gonna  
remember that far back? 

  Anyway, I thought I'd put in a really minor addition to your comments on 
the subject. They're excellent, but they also can be read in two ways;  
only one of which is correct. I'm sure you had the right one in mind when  
you wrote your note, but this may clear up any confusion among some who  
read it. 

  While a sphere, or any shape for that matter has capacitance, it does not 
require "the rest of the matter in the universe" in order for that 
capacitance to exist. In any first semester fields text, the capacitance of 
an isolated sphere is the first thing derived when they get to the subject of 
capacitance, because the math is so straightforeward. But in this derivation,  
no extra matter in the universe is assumed; in fact the implicit assumption 
is that there's absolutely no extra matter in existence. They usually say  
that there're no other bodies close enough to influence it, but without  
quantitatively defining what constitues "influence", they're really assuming 
that there's nothing else there, or perhaps that any other matter is  
infinitely far away. (Same difference, I guess.) 

  In any case, an arbitrary charge q on the body produces an electric field 
which can be predicted. The electric field is integrated from the surface of 
the sphere to a point infinitely far away, giving the potential (voltage) 
between the sphere and that point, and then the familiar old C = q/V is 
applied to get the capacitance between the sphere and the point infinitely 
far away. This is the equation we tend to (ab)use for a spherical capacitor. 

  Now, I don't expect that you implied that extra matter in the universe was 
required to greate a capacitance on the isolated sphere, but a quick reading  
of your note could give that impression. Since you immediately followed the 
portion I quoted with a comment that the equation for the capacitance of a 
sphere doesn't work perfectly in the Real World, I suspect you were referring 
to the fact that additional matter in the vicinity will affect the value of 
the sphere's capacitance, and the capacitance we measure won't match up to  
that nice, idealized equation, becaue we've violated the assumptions that 
made it so easy to derive.  

  Predicting the real capacitance in such a situation would take lots  
more work that just might not be worth the effort. Using a computer 
with decent number-crunching power along with the Method of Moments 
should yield pretty accurate results, but if the idealized equation 
is already within 20% of the actual as you seem to have measured,  
there would seem to be better things to do with our time. Do the  
equations for toroid capacitances come this close too? 

  Anyway, I thought I'd put in my two cent's worth. Hope that's okay. 

Wes B.