Solved Problem on Capacitors
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A capacitor is formed by two plates of area A and distance 2d, between these plates are placed two insulators of dielectric constants k1 and k2 and thickness d, as shown in the figure. Determine the capacitance of this new capacitor. Given the vacuum permittivity ε0.


Problem data:
  • Area of the plate of the capacitor:    A;
  • Distance between capacitor plates:    2d;
  • Dielectric constant of the insulation 1:    k1;
  • Thickness of insulator 1:    d;
  • Dielectric constant of the insulation 2:    k2;
  • Thickness of isulator 2:    d;
  • Vacuum permittivity:    ε0.
Solution

This capacitor behaves as two different capacitors in series, filled by insulators of dielectric constants k1 and k2 (Figure 1). The capacitance of a capacitor is given by
\[ \begin{gather} \bbox[#99CCFF,10px] {C=\varepsilon_0\frac{A}{d}} \end{gather} \]
Figure 1

If there were no presence of the insulator (if were made vacuum between the plates) the capacitances would be
\[ \begin{gather} C_{01}=C_{02}=\varepsilon_0\frac{A}{d} \tag{I} \end{gather} \]
With the presence of the insulator, the new capacitance will be given by
\[ \begin{gather} C_1=k_1 C_{01} \tag{II-a} \end{gather} \]
\[ \begin{gather} C_2=k_2 C_{02} \tag{II-b} \end{gather} \]
substituting the equation (I) into the equations (II-a) and (II-B)
\[ \begin{gather} C_1=k_1\varepsilon_0\frac{A}{d} \tag{III} \end{gather} \]
\[ \begin{gather} C_2=k_2\varepsilon_0\frac{A}{d} \tag{IV} \end{gather} \]
For a series of two capacitors, the equivalent capacitor Ceq is given by
\[ \begin{gather} \bbox[#99CCFF,10px] {C_{eq}=\frac{C_1 C_2}{C_1+C_2}} \tag{V} \end{gather} \]
substituting equations (III) and (IV) into equation (V)
\[ \begin{gather} C_{eq}=\frac{k_1\varepsilon_0\dfrac{A}{d}k_2\varepsilon_0\dfrac{A}{d}}{k_1\varepsilon_0\dfrac{A}{d}+k_2\varepsilon_0\dfrac{A}{d}} \end{gather} \]
factoring   \( \varepsilon_0\frac{A}{d} \)   in the denominator
\[ \begin{gather} C_{eq}=\frac{k_1\cancel{\varepsilon_0\dfrac{A}{d}}k_2\varepsilon_0\dfrac{A}{d}}{\cancel{\varepsilon_0\dfrac{A}{d}}\left(k_1+k_2\right)} \end{gather} \]
\[ \begin{gather} \bbox[#FFCCCC,10px] {C_{eq}=\frac{k_1 k_2}{k_1+k_2}\varepsilon_0\frac{A}{d}} \end{gather} \]
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Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .