This problem deals with a capacitive accelerometer structure shown below. The accelerometer consists of a cube of silicon $($side dimension, a$,$ of $1$mm$)$ that is supported using two beams as shown. Assume the following dimensions for the accelerometer:

$*$ The side of the proof mass cube, a$,$: $1$mm $*$ Length of each beam, L $=50$m

$*$ Width of each beam, w $=5$m $*$ Thickness of each beam, t $=5$m

$*$ Gap between proof mass and electrode, g $=5$m

$*$ Density of Si $=2330$ Kg$/$m$3$

$*$ Young's Modulus of Si : $190$ GPa $($all directions$)$

Calculate the sensitivity of this accelerometer to acceleration in z direction, $\backslash$Delta C$/\backslash$Delta a$.$