Question

2) Consider a coaxial capacitor with two concentric metal cylinders of radii a and b (a < b), filled with a dielectric material whose permittivity e varies linearly from ea at r = a to Ep at r = b. a. Find the capacitance per unit length. b. Find the numerical value of the capacitance if the radii are 2 mm and 6 mm and the relative dielectric constant varies from 2.25 to 8.5, respectively.

Answer 1

The electric field (E) inside the capacitor exists in only the dielectric materials and points readially inward. Let E be the field in dielectric Ea and Es be the field in dielectric E. At the interface between the two dielectric sections, E is parallel to E and both are tangential to the interface since boundary conditions roquire that the tangential components of Ei and Ea be the Same, it follows that E, = E2 = – FE

D E E 5 Ps 1 8 EE - SC ore Psi - - CE similarly in medium 2 P2 - ERE Thus the & fields will be the same in the two dielectrics, but the charge densities will be different along the two sides of the inner conducting cylinder. since the same voltage applies for the two sections of the capacitor, we can treeat them as two capacitons in parcalet Now C, Atal similarly = TEbe en bla) capacitance pen = al(Eat Eb) Iunit length X(Catch scoatl - XX (2.25 +8.5x8 185 x 1612 en (6/2) - 2.42 x 1010 F