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Find the capacitance of a system of thre...

Find the capacitance of a system of three parallel plates, each of area A `metre^(2)` separated by distances `d_(1) and d_(2)` metre respectively. The space between them is filled with dielectrics fo relatives dielectric constants `K_(1) and K_(2)`. The dielectric constant of free space is `in_(0)`.

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A system of three parallel plates is forming two capacitors of capacities `C_(1) and C_(2)` connected in series as shown in Fig.
`C_(1) = (K_(1) in_(0) A)/(d_(1)) and C_(2) = (K_(2) in_(0) A)/(d_(2))`
The equivalent capacity of the system is
`(1)/(C ) = (1)/(C_(1)) + (1)/(C_(2)) or C = (C_(1) C_(2))/(C_(1) + C_(2))`
`C = ((K_(1) (in_(0) A)/(d_(1))) (K_(2) (in_(0) A)/(d_(2))))/(in_(0)A((K_(1))/(d_(1)) + (K_(2))/(d_(2)))) = (K_(1) K_(2) A ((in_(0))/(d_(1))) ((in_(0))/(d_(2))) d_(1) d_(2))/(in_(0) A (K_(1) d_(2) + K_(2) d_(1))) = (K_(1) K_(2) in_(0))/(k_(1) d_(2) + K_(2) d_(1))`
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