Recent advances in microelectromechanical systems $($MEMS$)$ have allowed for the manufacture of extremely small sensors that can be placed within the human body. One such device, a smart stent, incorporates a pressure$-$sensitive $LC$ circuit, shown on the right below, with a stent, which is a tube placed within an artery to prevent its closure.

Mutual inductance coupling

The capacitance of the device is variable, and responds to changes in the pressure within the blood vessel: an increase in pressure moves the plates of the capacitor closer to one another. After the smart stent is placed within an artery, the blood pressure in the artery can be remotely monitored by inducing a current in the inductor of the $\frac{?}{{?}_C}$ circuit and using a receiver to record the resonance frequency of the circuit $($see the previous figure$).$ During testing in a patient, it is found that the capacitance of the variable capacitor changes linearly from $0\mathrm{.}206$ pF at a blood pressure of $37\mathrm{.}5$ mmHg

$)$ to $0\mathrm{.}247$ pF at a pressure of $135$ mmHg $.$

$($a$)$ What should the inductance of the inductor in the circuit be $($in $H)$ if the resonance frequency of the circuit is $203$ MHz at $135$ mmHg $?$

$$

$($b$)$ What is the resonance frequency $($in MHz $)$ of the circuit at $37\mathrm{.}5$ mmHg $?$

$$ MHz

$($c$)$ If the capacitor in the circuit has parallel square plates of side $4\mathrm{.}10$ mm $,$ what is the change in the distance between the plates $(inm)$ when the blood pressure in the artery increases from $37\mathrm{.}5$ mmHg to $135$ mmHg $?$

$m$