Solution:

$ \begin{aligned}\\ &g_n=\left.\frac{\delta I_0}{\delta V G s}\right|_{V o s-c o n s t a n t} \\\\ &\therefore g m=\mu_n \cos \frac{w}{L}(V G S-V r H)-(1) \\\\ &\text { also } g m=\sqrt{2 \ln \cos W / L \text { Id }} \\\\ &\therefore g m=\frac{2 I_D}{(V G S-V \Gamma H)}...(2) \\\\ & \end{aligned}\\ $

From (1) & (2)

$ \begin{aligned}\\ &\mu n \cos \frac{\omega}{L}\left(V_{G S}-V_{T H}\right)=\frac{2 I_D}{V_{G S S}-V i H} \\\\ &\therefore(V G S-V T H)^2=\frac{2 I_D}{\mu_n \cos \frac{W}{L}} \\\\ &\therefore \frac{W}{L}=\frac{2 I_D}{\mu_n \operatorname{Cos}\left(V_{G s}-V / H\right)^2}...(a) \\ \end{aligned} $

From (1)

$ \frac{W}{L}=\frac{9 m}{\ln (0 x(V G S-V r H)}-\text { (B) } $

a) When $I_D$ is constant,

b) Whengm is constant,