Question #43363

A solid conductor is shaped as a spherical sector (a truncated cone bounded by two spherical caps centered at the apex). The inner spherical cap has a radius $r_1 = R$ and the outer spherical cap has a radius $r_2 = 3R$. The solid angle subtended by the sector at the apex is $\Omega$. The conductor is made of a non-Ohmic material where the local electric field magnitude $E$ and current density magnitude $J$ are related by $E = \beta J^2$, where $\beta$ is a positive constant. A steady current $I$ flows radially outward from the inner cap to the outer cap. The conductor is conceptually divided into two segments by an intermediate spherical cap of radius $r = 2R$. What is the ratio of the total power dissipated in the inner segment ($R \le r \le 2R$) to that in the outer segment ($2R \le r \le 3R$)?