= Long-channel MOSFET current law
{title2=$I_D=\mu C_{\rm ox}(W/L)[(V_{GS}-V_T)V_{DS}-V_{DS}^2/2]$}
For an n-channel <MOSFET>, assume a gradual channel, constant <charge carrier mobility> $\mu$, gate capacitance per area $C_{\rm ox}$, width $W$ and length $L$. With overdrive $V_{\rm ov}=V_{GS}-V_T>0$, the inversion-sheet charge magnitude is $C_{\rm ox}(V_{\rm ov}-V(x))$. The drift-current relation $I_D=\mu WC_{\rm ox}(V_{\rm ov}-V)V'$ is constant along the channel. Integrating from $V=0$ to $V_{DS}$ over length $L$ proves the displayed expression for $0\leq V_{DS}\leq V_{\rm ov}$. At the drain pinch-off condition $V_{DS}=V_{\rm ov}$, it yields $I_{D,\rm sat}=\mu C_{\rm ox}WV_{\rm ov}^2/(2L)$. This ideal saturation neglects channel-length modulation and short-channel velocity saturation.
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