Exercise:
The distance between the conductors of a capacitor is often small compared to the area of the surfaces. In these situations the capacitance of a parallel plate capacitor is a good approximation. abcliste abc Show that the capacitance of a spherical capacitor with a small distance between the spherical shells can be approximated by a parallel plate capacitor with the same distance and an area that corresponds to the surface area. abc Show that the capacitance of a cylindrical capacitor with a small distance between the cylindrical surfaces can be approximated by a parallel plate capacitor with the same distance and an area that corresponds to the mantle area. abcliste

Solution:
abcliste abc The radii of the concentric shells are r and r+d. It follows for the capacitance C pivarepsilon_leftfracr-fracr+dright^- pivarepsilon_leftfracr+d-rrr+dright^- pivarepsilon_fracrr+dd For d ll r the distance between shells can be neglected compared to the radius i.e. r+dapprox r. It follows that C &approx pivarepsilon_fracr^d varepsilon_fracpi r^d varepsilon_fracAd with Api r^ i.e. the surface of a spherical shell with radius r. The graph shows the exact capacitance and the approximation as a function of the ratio d/r and the ratio of the two quantities right axis. center includegraphicswidthtextwidth#image_path:graph-spherical-capacitor# center abc The radii of the concentric cylinders are r and r+d. It follows for the capacitance C fracpivarepsilon_ Llnfracr+dr fracpivarepsilon_ Lln+d/r For small values of x the logarithm of +x can be approximated by x. This leads to C &approx fracpivarepsilon_ Ld/r varepsilon_fracpi r Ld varepsilon_fracAd with Api r L i.e. the mantle area of a cylinder with length L and radius r. The graph shows the exact capacitance and the approximation as a function of the ratio d/r and the ratio of the two quantities right axis. center includegraphicswidthtextwidth#image_path:graph-cylindrical-capacitor# center abcliste