A silicon$-$based micro cantilever with the following dimensions:

Thickness: $2m,$

Width: $20m,$

Length: $39\mathrm{.}4m$

Calculate the minimum detectable force using the designed cantilever $($assuming only Johnson noise is present, the operating temperature is $300$ K $,$ and the bandwidth of the measurement circuit is $1$ kHz $).$ Additionally, calculate the minimum detectable force when the output voltage of the WB circuit is less than $1V$ due to additional noise effects. Assume the cantilever material is silicon $($with a Young's modulus of $155$ GPa $),$ and the strain sensor material is aluminum $($with a resistivity of $3{10}^{-8}*m),$ with a strain gauge thickness of $100$ nm $($assumed to have no effect on the effective spring constant of the silicon cantilever$).$ In the calculation of resistance change in aluminum due to the applied force, assume no change in the width and thickness of the gauge. For the calculation of the tensile stress due to the applied force, refer to the method outlined in the e$-$class announcement $($note that stress varies along the length of the cantilever when a force is applied to the free end$).$