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Why does the capacitance of two cylindrical capacitors of same length stay the same if the ratio of the outer radii to the inner radii of one capacitor is same as the other.

The capacitance of a cylindrical capacitor is C = (2πel)/(ln(R2/R1)) where e - epsilon symbol, l - length of the capacitor, R2 and R1 are the outer and inner radius respectively.

According to the equation its pretty clear but I want an explanation that's more intuitive, one that does not need equations for an explanation.

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  • $\begingroup$ 1) Can you post the equation you are referring to? 2) I think you mean you are looking for an explanation that is more intuitive, if you want an answer that doesn't involve equations. 'Analytical' is the opposite - it usually means involving equations (i.e. mathematical analysis). $\endgroup$
    – Time4Tea
    Commented Aug 20, 2018 at 17:41
  • $\begingroup$ @Time4Tea Oh yeah. There goes my poor english. The equation was C = (2πel)/(ln(R2/R1)) where e - epsilon symbol (I couldn't find the actual symbol), l - length of the cylinder and R2 and R1 are the outer and inner radii respectively. $\endgroup$
    – Abha Vishwakarma
    Commented Aug 20, 2018 at 18:47
  • $\begingroup$ Ok. Would be best if you could edit and add it to the question. $\endgroup$
    – Time4Tea
    Commented Aug 20, 2018 at 18:50
  • $\begingroup$ @Time4Tea Oh my god I am so sorry I just realised I didn't write the question properly. Thanks for pointing it out. $\endgroup$
    – Abha Vishwakarma
    Commented Aug 20, 2018 at 18:51
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    $\begingroup$ @Time4Tea Please read the question once again. I edited it and now I think I have made my question clear to understand. $\endgroup$
    – Abha Vishwakarma
    Commented Aug 20, 2018 at 18:55

1 Answer 1

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The capacitance of a capacitor, in general, is proportional to the plate area and inversely proportional to the distance between the plates. So, if we keep the ratio between the area of the plates and the distance between the plates the same, the capacitance should remain the same.

If the ratio of the inner and outer radii of a cylindrical capacitor stay the same, it means that both radii have changed by the same factor, which means that the difference between the radii has changed by the same factor and the areas of the plates (since the area of a cylinder is proportional to its radius) have changed by the same factor.

So, since, the ratio between the area of the plates and the distance between the plates have not changed (because both have changed by the same factor), we can conclude, according to our initial observation, that the capacitance should not change either.

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  • $\begingroup$ How is the difference between the radii changing by the same factor? For example suppose I have a cylindrical capacitor 'A' with ratio of outer to inner radii = 6/2 = 3 and another cylindrical capacitor 'B' with ratio 9/3 = 3. Their ratio's are same but difference in the radii is 4 for 'A' and 6 for 'B'. They aren't changing by the same factor. Moreover a cylindrical capacitor is not a parallel plate capacitor so the comparison between the two needs further justification. $\endgroup$
    – Abha Vishwakarma
    Commented Aug 20, 2018 at 17:10
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    $\begingroup$ @AbhaVishwakarma in your example, the ratio of the two radiuses is staying the same - both are increasing by 3/2 and you have shown that their ratio is constant (3). You seem to be confusing multiplication and addition. $\endgroup$
    – Time4Tea
    Commented Aug 20, 2018 at 17:45
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    $\begingroup$ @AbhaVishwakarma Re. the second part of your comment (the first part has already been nicely addressed by Time4Tea), the analogy btw a parallel plate capacitor and a cylindrical capacitor is more convincing, when the difference between radii is small (field is close to uniform) and the cylindrical capacitor is viewed as a number of parallel plate capacitors connected in parallel. When the diff-ce in radii is not small, we can view the cyl. capacitor as a number of cyl. capacitors in series, formed by a number of concentric cylinders, and use the same approximation for each pair of cylinders. $\endgroup$
    – V.F.
    Commented Aug 20, 2018 at 18:34

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