Using Capacitors to Measure Fluid Level

Measuring the level of a fluid is useful for a variety of applications, and the technology for the techniques employed in the acquisition of this measurement has progressed far beyond the use of sight glasses and mechanical floats. In fact, a widely used method to measure the amount of fuel in a gas tank is with a device that floats on top of the fuel combined with a sensor, the fuel gauge sending unit, that translates the angle of the float to the amount of fluid in the tank. With this method, the gauge tends to change position with the angle of the car as well as the angle of the float relative to the fluid, so a lot of the time the gauge position can be misleading. A more modern technique of measuring fluid level involves capacitors, and this article will detail the concepts and mathematics behind the relationship with fluid height.

Conceptual Background

Capacitors

For more details, see capacitor.

A parallel plate capacitor with conductive plates of area [math]\displaystyle{ A }[/math] separated by a dielectric with a gap distance of [math]\displaystyle{ d }[/math].

A capacitor consists of two conductors, e.g. conducting plates, separated by some kind of insulator. The insulator--a dielectric--within the gap between the two conductors can be air, plastic, glass, etc. Additionally, the conductors have to be connected to some sort of power supply in order to acquire a buildup of charge on the surface of the conductors.

The magnitude of the electric field within a capacitor is [math]\displaystyle{ \left| \vec{E}_{capacitor} \right| = \frac{Q/A}{\epsilon_{0}} }[/math], where the gap is only occupied by free space. [math]\displaystyle{ Q }[/math] is the charge on a plate and [math]\displaystyle{ A }[/math] is the area of a plate. Additionally, [math]\displaystyle{ \epsilon_{0} }[/math] is the permittivity of free space as detailed here.

With a dielectric involved, we apply the equation [math]\displaystyle{ \vec{E}_{dielectric} = \frac{\vec{E}_{applied}}{K} }[/math], where [math]\displaystyle{ K }[/math] is the dielectric constant of the material, and get [math]\displaystyle{ \left| \vec{E}_{dielectric} \right| = \frac{Q/A}{K\epsilon_{0}} }[/math]. The capacitance, [math]\displaystyle{ C=\frac{Q}{V} }[/math] (where [math]\displaystyle{ V }[/math] is the voltage) or [math]\displaystyle{ C=\frac{\epsilon_0 A K}{d} }[/math], therefore changes with a varying dielectric constant.

Relating Capacitance to Fluid Level

When multiple materials are between the gap, for instance water and air, the overall capacitance would be [math]\displaystyle{ C=\epsilon_0 A (\frac{K_{water}}{d_{water}}+\frac{K_{air}}{d_{air}}) }[/math], where [math]\displaystyle{ d=d_{water}+d_{air} }[/math] and [math]\displaystyle{ K_{air} \approx 1 }[/math].

A Computational Model

Error creating thumbnail: sh: /usr/bin/convert: No such file or directory Error code: 127

A Python simulation depicting how varying the amount of dielectric within the gap of a capacitor (with two oppositely charged conducting plates) changes the corresponding electric field between the gap. Notice how more of a dielectric with a high dielectric constant reduces the electric field, which in turn increases the capacitance.

For a simple simulation depicting how changing the amount of dielectric within the gap of a capacitor changes the corresponding electric field, please check out the VPython simulation. The charge on the capacitor as well as the dielectric constants for both dielectrics can be adjusted to visualize how the changes affect the electric field. Notice that the scale factor for the electric field is computed in such a way that the maximum length of the arrow is always the distance between the two plates, so adjusting constants proportionally may result in similar looking arrows (although the electric field will be different).

Examples

Be sure to show all steps in your solution and include diagrams whenever possible

Connectedness

1. How is this topic connected to something that you are interested in?
2. How is it connected to your major?
3. Is there an interesting industrial application?

History

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

See also

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

Further reading

Books, Articles or other print media on this topic

External links

[1]

References

This section contains the the references you used while writing this page