Question #89433

A semi-infinite distributed capacitive network is constructed along the $x$-axis from $x = L$ to $x \to \infty$. The network is filled with a non-uniform dielectric material. For any small segment of length $dx$ at position $x$, the series capacitance is given by $C_s = \frac{\epsilon_0 k_s(x) A}{dx}$ and the shunt capacitance to ground is $C_p = \frac{\epsilon_0 k_p(x) w}{d} dx$, where $A, w,$ and $d$ are geometric constants. The dielectric constants vary spatially as $k_s(x) = k_0 \left(\frac{x}{L}\right)^2$ and $k_p(x) = k_0 \lambda$, where $k_0$ and $\lambda$ are positive dimensionless constants. Given the geometric relation $\frac{w}{d} = \frac{A}{L^2}$, and defining a characteristic capacitance $C_0 = \frac{\epsilon_0 k_0 A}{L}$, what is the exact equivalent capacitance of the entire network as seen from the input terminal at $x = L$?