A MOSFET is designed to be used as an amplifier in part of a magnetic sensor circuit, which is subjected to a perpendicular magnetic field in an environment. You are tasked with determining the impact of this magnetic field on the current flow through the MOSFET and the corresponding change in its transfer characteristics.
Parameters are provided below: Vth =1V, L=5μm, W=20μm, Cox=100nF/cm^2, μ_0=1000 cm^2/Vs, μn is calculated using the zero-field electron mobility as μn = μ0/(1+(μ_0*B)^2).

1. Assume B=0.5T as a constant, and find the expression of the change of drain current ΔId due to this magnetic field. (1)

2. Assume B and Vgs are time-variant parameters with the same function f(t), i. e. B^2 = ln(f(t)) and Vgs = f(t).
(1) Derive the expression of ΔId in terms of Vgs.(1.5)
(2) Plot Id vs.Vgs with and without the magnetic field. How ΔId change when Vgs increases? (1)
(3) Take the derivative of ΔId with respect to Vgs, does this comply with what you have found in the previous part? (1.5)