$($Somehow solve it please whatever you can do$.$its urgent $!)$

Current density in an interconnect, for a harmonic voltage $V(t)=V_0{e}^{jt},$ is given by$,$

vec$(J)(z)=J_0{e}^{-\frac{(j+1)z}{}}$hat$(x)$

where,

$=\sqrt[\phantom{2}]{\frac{2}{}}$

and $$ is the conductivity. $J_0$ is the D$.$C$.$ value of current density, $J_0=\frac{V_0}{l}.,l$ is the length of the interconnect.

If the resistance of the interconnect $($see figure above$),$ per unit length is defined as$,$

$R=\frac{1}{l}$Real$\{\frac{V_0}{\text{I}}\}$

I being the total current flowing through the interconnect, then,

$($a$)$ Derive the expression for $R.$

$($b$)$ What is the expression for $R$ in the limits

$($i$)b,($or $\frac{b}{}$

$($ii$)$ or $$

Let your expression be in terms of $a,b,,.$ Comment on the physical significance of the results in the two cases.

$($c$)$ Consider the case when $$ is very small, smaller than even the surface roughness. The random nature of surface roughness is characterized by two parameters $($a$)Z_r,$ the RMS value of variation in surface height from the mean; and $($b$)$ The Auto$-$correlation function $F_{}(x)=Z_r^2{e}^{-\frac{x^2}{x_z^2}},$ where $x_c$ is the correlation length. The Auto$-$correlation function characterizes the sharpness $($spatial frequency$)$ of the surface roughness, while the $Z_r$ characterizes the magnitude of the variation. Under the assumption that $$ is so small that the current will flow very close to the surface and follow the profile of surface roughness, can you derive the effective length $($which will be greater than physical length $l)$ of the interconnect? Justify your approach. Meaningful attempts will get partial marks. There is a possibility of getting additional $+3$ bonus marks for exceptional answers. Even if you solve it after the exam, you may meet me to discuss.