What is the limit I have to take so that the capacitance formula for concentric cylinders shows the capacitance of infinite plates per surface area?

Question

I'm trying to figure out how to solve this problem. The formula for the capacitance of two concentric cylinders is given as \$C=2\pi\varepsilon L/ln(b/a)\$, where b is the outer radius, a is the inner radius and L is the length.

So I guess the goal is to prove that by taking a limit, it converges to \$C=\varepsilon \frac{A}{d}\$, where d is the distance between them, or b-a, and A is the area of the plates.

What I've thought is taking the limit of a and b as they approach infinity, i.e. the cylinders expand to infinite perimeters, but I couldn't find anything.

Thanks in advance.

Answer 1

From Wikipeida, $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots $$ where \$|x| < 1\$.

For very large values of the cylinder diameter and small gap (\$d = b-a\$) between them,

$$\ln(\frac{b}{a}) = \ln(\frac{a+d}{a})=\ln(1+\frac{d}{a}) = \frac{d}{a} - \dots$$

\$|d/a| < 1\$ for large diameter cylinder with small gap between plates.

Neglecting the higher powers, the formula given in the question for cylinder becomes

$$C \approx 2\pi \varepsilon L / (d/a) = 2\pi \varepsilon L \cdot a/ d$$

Note that

1. \$d\$ appears in the denominator as in target formula of parallel plates.
2. \$\varepsilon\$ appears in the numerator as in target formula.
3. \$2 \pi a \approx 2\pi b\$ is circumference of the inner / outer pipe.
4. \$2 \pi a L\$ is an area similar to that in the target formula.

I couldn't figure out how to eliminate the 2piL part.

With the above shown approximation of \$\ln()\$, it is clear that you need to group \$2\pi\$ with \$a\$ rather than \$L\$.

I have probably not gotten the actual concept of the problem in the first place.

The "vertical" direction of parallel plates correspond to L for the cylinder. The "horizontal" direction of the parallel plates correspond to the circumferential direction of the cylinder. A way to visualise this is that you "cut" the cylinder(s) along the length and unroll it to get flat rectangular plate(s). This rectangular plate(s) then look similar to the rectangular plates in the parallel plate capacitor.