Dielectric effect in a Cilindric Capacitor

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Description of the Experiment

Figure 1: Photo of the experimental setup

This experiment has the purpose of determining the capacity of a variable coax cylindrical capacitor. It has two sections, a first with a PVC dielectric and a second with air, allowing the determination of the relative dielectric constant of PVC.


Ligações

  • Video: rtsp://elabmc.ist.utl.pt/condensador.sdp
  • Laboratory: Advanced in e-lab.ist.eu[1]
  • Control room: Condensador Cilíndrico
  • Level: ****


Experimental Apparatus

Figure 2: Sectioned schematic of the capacitor, where a=12mm and b=14mm

The capacitor is made from a copper tube with 12mm outer diameter and 200mm length, having been outfitted with a PVC dielectric with 100mm length. This is inside a second copper tube with 14mm inner diameter that moves along the first, thus acting as the plates of a variable capacitor. This way, the area of the capacitor corresponds only to the parts where the two cylinders coincide.

This configuration corresponds to two capacitors whose total capacity is given by the shunt between this system and a fixed capacitor (initial capacity). The experiments outputs the frequency at which the RC circuit oscillates (R = \( 1.8 M \Omega \)). The period of the oscillating circuit is given by \( T= \frac{RC}{1,44} \).


Protocol

Determination of the relative dielectric constant

Take two sets of experimental data, one covering the dielectric area and the other covering air. The end-points of the sweep should be chosen in a way that allows a precise determination of the slope of the graphical representation. This slope will give the relation between the capacitor and the oscillation period (each set referring to its dielectric). The ration between the two slopes will give the relative dielectric constant for PVC.


Advanced Protocol

Determination of the dielectric constant of air

Considering Gauss's law, it is possible to determine the capacity of the cylindrical capacitor, using the formula:

\[ C = 2 \times \pi \times \epsilon _0 \times \frac{L}{ ln(\frac{b}{a}) } \]

By doing a linear regression with the data from the section of the capacitor without the dielectric (which means the air acts as the dielectric) it is possible to accurately determine the value of the capacity. From this, reversing the formula, the air's permittivity can be found (close to vacuum, i.e. \( \epsilon _0 \)).


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