Consider a cylindrical capacitor, with outer radius \(R\) and cylinder separation \(d\). Determine what the capacitance approaches in the limit where \(d \ll R\). (Hint: Express the capacitance in terms of the ratio \(d / R\) and then examine what happens as the ratio \(d / R\) becomes very small compared to \(1 .)\) Explain why the limit on the capacitance makes sense.

To find the capacitance of a cylindrical capacitor, we can use the following formula: C = \frac{2 \pi * \epsilon * L}{ln (\frac{R}{r})} where C is the capacitance, ε is the permittivity of the dielectric, L is the length of the capacitor, R is the outer radius, and r is the inner radius. In our problem, we are given the outer radius (R) and the cylinder separation (d) instead of r. We can find r using the given information: r = R - d Now, let's plug this into the capacitance formula: C = \frac{2 \pi * \epsilon * L}{ln (\frac{R}{R-d})}

To express the capacitance in terms of the ratio d/R, we need to rewrite our formula for r, the inner radius, in terms of this ratio. We can set up a proportion as follows: \frac{d}{R} = \frac{R-r}{R} So, r = R(1-\frac{d}{R}) Now, we can substitute this into our capacitance formula: C = \frac{2 \pi * \epsilon * L}{ln (\frac{R}{R(1-\frac{d}{R})})} Simplifying the expression: C = \frac{2 \pi * \epsilon * L}{ln (\frac{1}{1-\frac{d}{R}})}

As the ratio d/R approaches 0, we can observe the behavior of the capacitance: \lim_{\frac{d}{R} \rightarrow 0} C = \frac{2 \pi * \epsilon * L}{ln (\frac{1}{1-0})} The limit simplifies down to: \lim_{\frac{d}{R} \rightarrow 0} C = \frac{2 \pi * \epsilon * L}{ln (1)} Since the natural logarithm of 1 is 0, the limit equals: \lim_{\frac{d}{R} \rightarrow 0} C = C_{max} = \infty Thus, as the cylinder separation d becomes very small compared to the outer radius R, the capacitance approaches infinity.

The limit we found, C approaching infinity as d/R approaches 0, makes sense for the following reasons: 1) Smaller cylinder separation (d) implies that the charges on the inner and outer cylinders are closer together, which creates a stronger electric field between them. 2) A stronger electric field implies a higher potential difference between the capacitor plates, and thus an increased capacitance. 3) In the limit where d/R approaches 0, the capacitance becomes infinitely large, reflecting the fact that an infinitely small separation would result in an infinitely strong electric field and infinite potential difference. Hence, our obtained limit on the capacitance is logically consistent and makes sense in the context of a cylindrical capacitor.