Hello again,
Let me illustrate the slanted plate capacitor and you can take it from there. Note that the dimensions are outlined in my previous post attachement, the capacitor labeled "Cap 2".
We start with the parallel plate capacitor formula, which is:
Cp=Ke0*A/D
where
Cp is the capacitance,
Ke0 is er*e0,
A is area,
D is distance between plates,
er is dielectric constant,
e0 is free space permittivity.
We want to break the slanted cap down into smaller sections and calculate each section separately, then add the results. If we break them down into infinitesimally small sections, we get an exact expression for the capacitance in the end although we have to remember that these techniques are only approximations to begin with just like the parallel plate capacitor.
Each small cap will have capacitance:
Ck=Ke0*Ak/Dk
where
Ak is the incremental area, and
Dk is the incremental distance which changes over the width x.
We'll declare the increment in x (usually called 'delta x') as 'dx', and define everything in terms of dx or x alone. When we let dx approach zero we get an exact expression because as dx goes to zero the error in the calculation of one Ck goes to zero.
The actual width is slanted and so has a slope M, and is related to dx by:
Wk=sqrt(dx^2+(M*dx)^2)
where M=H/W (H is the height of the plate alone and W is the width),
and this is simply the Pythagorean theorem applied to the base and rise, and simplifying we get:
Wk=dx*sqrt(M^2+1)
and all this means is that for any increment (dx) in the left to right distance (x) we have a plate width Wk.
To find the area Ak now all we have to do is multiply Wk times the length of the plate (the length of the plate is the distance across the plate looking into the page) and we get:
Ak=Wk*L
where
L is the length.
Now that we know what the plate width is relative to dx, we then write an expression for Dk the distance as that changes with x:
Dk=2*M*x+D1
where D1 is the same as H1, and H1 is the left hand side plate separation, and note this is just the equation for a line y=m*x+b applied to the two slants.
Now recalling the incremental capacitance formula:
Ck=Ke0*Ak/Dk
and substituting Ak and Dk with the expressions we formed above, we collectively end up with:
Ck=dx*Ke0*L*sqrt(M^2+1)/(2*x*M+D1)
Note that at this point if we call dx "delta x" and make dx small (but not zero) and sum all the Ck we would get an approximation:
Sum (Ck)=Sum (dx*Ke0*L*sqrt(M^2+1)/(2*x*M+D1))
so if we let dx=W/10 for example (break the cap up into 10 smaller caps) we would get a rough approximation although there would be a small error in the calculation of the total C.
But taking the limit as dx approaches zero we get an exact expression:
C=integral[0 to W](Ke0*L*sqrt(M^2+1)/(2*x*M+D1)) dx
where W is the total width of the capacitor plates.
After computing that integral, whatever that might take, we end up with a nice little expression:
C=(Ke0*L*sqrt(M^2+1)*(ln(2*M*W+D1)-ln(D1)))/(2*M)
As a basic sanity check, if we make the plate height H zero we end up with a parallel plate capacitor again, so if we take the limit of the above expression as the height H approaches zero we should get the same expression back again as for the parallel plate capacitor:
Cp=lim[H-->0]{ (Ke0*L*sqrt(M^2+1)*(ln(2*M*W+D1)-ln(D1)))/(2*M)}
and this is what we get:
Cp=(Ke0*L*W)/D1
which is indeed the formula for the parallel plate capacitor with A=L*W and D=D1.