Five semicircular conducting plates, each with radius $a,$ are attached to a conducting rod along their common axis. The plates are parallel with a separation gap $g$ between adjacent plates. A similar arrangement involving five additional semicircular conducting plates is attached to a second conducting rod symmetrically intertwined, upside down, with the first arrangement allowing the plates to rotate freely, as shown below.

The two five$-$plate constructions, with an adjacent plate separation of $\frac{g}{2},$ are insulated from each other, and wire leads are attached to the separate axles. This is a common type of variable capacitor, with a capacitance proportional to the overlap angle $.$

$($a$)$ Show that the net capacitance can be written as

$C()=\frac{5lo{n}_0{a}^2}{g}$

$($b$)$ The plate radius is a$=5\mathrm{.}9$cm while the gap distance is g$=2\mathrm{.}1$mm$.$ This variable capacitor is placed in series with a $120$ Microhenries inductor and a $10$ ohms resistor. An amplified signal from an antenna supplies a voltage source, and the output voltage is taken across the capacitor. The circuit is used as a radio receiver that receives a signal frequency equal to the resonance frequency of the L$-$R$-$C circuit. What capacitance should be used to receive an AM radio signal at $1250$ kHz$?$ What angle theta $($in degrees$)$ should be selected to give that capacitance?

c$)$ Draw a phasor diagram for this circuit.

d$)$What are the reactances of each component.

e$)$If the input signal is a sinusoidal voltage with an amplitude of $100$mV $($at the AM frequency above$),$ what is the amplitude of the output voltage.

f$)$ If the knob on the capacitor is now turned so that theta$=120$ degrees, what is the frequency the radio is now turned to$?$