Three identical parallel plate capacitor (air in between the plates) C₁, C₂, C₃ have capacitance C each. The space between their plates is now filled with dielectrics as shown. If all three capacitors still have equal capacitance, obtain the relation between dielectric constants k, k₁, k₂, k₃ and k₄.

Solution

Given, C₁,C₂ and C₃ are three identical parallel plate capacitor with capacitances C each.
Initially, air is filled in between the parallel plates as the medium.
After introducing the dielectrics, the capacitance of capacitors C₁, C₂ and C₃respectively is given as,

Capacitance across first capacitor, C_{1} = \frac{{kε}_{0} A}{d} Capacitance across second capacitor, C_{2} = \frac{2 ε_{0} A}{d} [\frac{k_{1} k_{2}}{k_{1} + k_{2}}] Capacitance across third capacitor, C_{3} = \frac{ε_{0} A}{2 d} [k_{3} + k_{4}]

Since, C₁=C₂=C₃we have,

\frac{{kε}_{0} A}{k} = \frac{2 ε_{0} A}{d} (\frac{k_{1} k_{2}}{k_{1} + k_{2}}) = \frac{ε_{0} A}{2 d} (k_{3} + k_{4})

k = \frac{2 k_{1} k_{2}}{k_{1} + k_{2}} = \frac{k_{3} + k_{4}}{2}

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