Electrostatics : Dielectrics in Capacitor

Question: Find the capacitance of the three parallel plates each of area A meter² and seperated by d₁ and d₂ meter. The in between spaces are filled with dielectrics of relative permittivity ε₁ and ε₂. The permittivity of the free space is ε₀

Solution:

The system is equivalent to two parallel plate capacitors connected in series . The spaces between the plates of the capacitors are filled with dielectrics media whose dielectrics constants K are ε₁ and ε₂. The dielectrics sonstant are also called relative permittivity.

Therefore the capacitances of two capacitors are

$C_1=\frac{{\epsilon }_1{\epsilon }_{0}A}{d_1}$

$C_2=\frac{{\epsilon }_2{\epsilon }_{0}A}{d_2}$

If C is the equivalent capacitance of the system,then

$\frac{1}{C}=\frac{1}{C_1}+\frac{1}{C_2}$

or

$C=\frac{{\epsilon }_0{\epsilon }_1{\epsilon }_{2}A}{{\epsilon }_2{d}_1+{\epsilon }_1{d}_2}$