Provide all the intermediate steps and explanations

$($a$)$ Calculate the total charge stored in the channel of an NMOS transistor if $C_{ox}=10f\frac{F}{}{m}^2,W=5m,L=0\mathrm{.}1m,$ and $V_{GS}-V_{tn}=1V.$ Assume $V_{DS}=0.$

$($b$)$ Derive the potential distribution along the channel of NMOS transistor. You need to integrate Eq$.()$ and hence solve for $V(x).$ There will be two solutions, which one is a valid solution and why? Also, sketch the behavior of $V^{}(x)$ for the case of onset of saturation.

$I_D=W{C}_{ox}[V_{GS}-V(x)-V_{tn}]{}_n\frac{dV(x)}{dx}$

Take ${}_n{C}_{ox}=200\frac{A}{V^2}$ and $V_{tn}=0\mathrm{.}4V.$

$($a$)$ An NMOS device carries current of $1$ mA with $V_{GS}-V_{tn}=0\mathrm{.}6V$; and $1\mathrm{.}6$ mA with $V_{GS}-V_{tn}=0\mathrm{.}8V.$ If the device operates in the triode region, calculate $V_{DS}$ and $\frac{W}{L}.$

$($b$)$ In the given circuit, M$1$ serves as an electronic switch. If is small, but not zero$),$ determine $\frac{W}{L}$ such that the circuit attenuates the signal by only $5\%.$ Assume $V_G=1\mathrm{.}8V$ and $R_L=100.$

Take ${}_n{C}_{ox}=200\frac{A}{V^2}$ and $V_{tn}=0\mathrm{.}4V,V_{tp}=-0\mathrm{.}4V.$

$($a$)$ Determine the region of operation of M$1$ in each of the circuits shown.give reason