Example 26.1 explored a cylindrical capacitor of length ℓwith radii a and b for the two conducto

step 1 So we have a cylindrical capacitor with a capacitance L over 2K, LN, B over A. So if we increase

Example 26.1 explored a cylindrical capacitor of length...

Example 26.1 explored a cylindrical capacitor of length $\ell$ with radii $a$ and $b$ for the two conductors. In the What If? section of that example, it was claimed that increasing $\ell$ by 10$\%$ is more effective in terms of increasing the capacitance than increasing $a$ by 10$\%$ if $b > 2.85 a$ . Verify this claim mathematically.

Key Concepts

Capacitance of Cylindrical Capacitors

This concept deals with the formulation of capacitance for a cylindrical capacitor, which typically involves the dimensions of the capacitor (its length and the radii of the two cylindrical conductors). The standard formula, C = (2????)/ln(b/a), shows that the capacitance is directly proportional to the length and inversely proportional to the logarithm of the ratio of the outer to inner radii. Understanding this relationship is fundamental to analyzing how changes in the physical dimensions affect the capacitor’s ability to store charge.

Logarithmic Dependence in Capacitance

The capacitance formula includes the natural logarithm of the ratio of the outer and inner radii, ln(b/a). This logarithmic term means that changes in the geometry do not affect the capacitance linearly. Instead, a percentage change in the radii leads to a change in the logarithmic term that is less pronounced for larger values of b/a. This dependence is crucial for determining the relative impact of varying the inner radius a compared to varying the length ?.

Sensitivity Analysis Using Differential Approximations

This involves using derivatives or finite difference methods to study how small percentage changes in the dimensions affect the overall capacitance. By comparing the fractional change in capacitance resulting from a percentage increase in the length ? versus the fractional change due to a percentage increase in the inner radius a, one can mathematically determine which parameter is more effective in increasing capacitance. Such analysis shows that for certain ratios of b to a (specifically when b exceeds about 2.85a), a 10% increase in ? yields a greater increase in capacitance than a 10% increase in a.

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