Design a silicon$-$based micro cantilever with a half$-$bridge circuit $($Two variable resistors are located on the upper surface of the cantilever, and their resistance increases as the force applied to the free end of the cantilever increases, while the other two resistors remain unchanged. The operating voltage of the Wheatstone Bridge is between $1$ and $3$ V $).$ The spring constant of the cantilever device required for application in an atomic force microscope should be $10\frac{N}{m}($Determine the thickness, width, and length based on this constant$).$

Derive the relationship between the force applied to the free end of the designed micro cantilever and the output voltage of the Wheatstone Bridge $($WB$)$ circuit, and plot a graph showing the result $($with the force on the $x-$axis and the output voltage of the WB on the $y-$axis$).$ Explain the meaning of each step in the derivation.

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$).$

Calculate the resonant frequency of the designed cantilever and explain the physical significance of this value from the perspective of sensor application.