Question

The space between the two coaxial cylindrical shells of radius $R_1$ and $R_2(R_1<R_2)$ is filled by a dielectric substances, whose dielectric constant varies with the distance from the axis as $k=A/r^3$. If the length of the cylindrical shells is '$\ell$' then find the capacitance of the system.

Answer 1

$k\oint E.ds=\frac{q}{{\in }_0}$
$\therefore E=\frac{q{r}^2}{{\in }_{0}A\left(2\pi \ell \right)}$
or, $E=\frac{q}{2\pi {\in }_{0}A\ell }\left(r^2\right)$
or, $V=-\stackrel{R_2}{\underset{\infty }{\int }}E.dr-\stackrel{R_1}{\underset{R_2}{\int }}E.dr=0+\stackrel{R_2}{\underset{R_3}{\int }}E.dr=\stackrel{R_2}{\underset{R_1}{\int }}\frac{q}{\left(2\pi {\in }_{0}A\ell \right)}{r}^{2}dr=\frac{q}{2\pi {\in }_{0}A\ell }\left[\frac{R_2^3-R_1^3}{3}\right]$
or, $V=\frac{q}{6\pi {\in }_{0}A\ell }(R_2^3-R_1^3)$
$C=\frac{q}{V}=\frac{6\pi {\in }_{0}A\ell }{(R_2^3-R_1^3)}$