Capacitive level sensor for liquids

[fuzzytoad]

Homework Statement

The quantity of liquid available in its storage tank is often monitored by a capacitive level sensor. This sensor is a vertically aligned cylindrical capacitor with outer and inner conductor radii Ra and Rb, whose length L spans the height of the tank. When a nonconducting liquid fills the tank to a height (h less than or equal to L) from the tank's bottom, the dielectric in the lower and upper region between the cylindrical conductors is the liquid (Kliq) and its vapor (Kv) respectively.

a) Determine a formula for the fraction F of the tank filled by liquid in terms of the level-sensor capacitance C
b) By connecting a capacitance-measuring instrument to the level sensor, F can be monitored. If L = 2.0m, Ra = 5.0mm, Rb = 4.5mm, Kliq = 1.4 and Kv = 1.0, what values of C (in pF) will correspond to the tank being completely full and completely empty?

Homework Equations

Capacitance for a cylinder: C = (2piεL)/ln(Ra/Rb)

and ε= Kε0

C=QV, Q is charge on the capacitor and V is potential difference between them

The Attempt at a Solution

I figured to treat the capacitor as 2 different cylindrical capacitors in parallel with different dielectrics in them.

C = Cliq + Cv
= (2piKliqε0h)/ln(Ra/Rb) + (2piKvε0(L-h)/ln(Ra/Rb)
after some simplification:
= (2piε0(hKliq - hKv + LKv)/ln(Ra/Rb)

I don't know how to get from an equation with h and L to F, since F is h/L.

 

[ideasrule]

What if you divide both sides by L?