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Three identical parallel plate (air) cap...

Three identical parallel plate (air) capacitors `C_(1), C_(2), C_(3) ` have capacitances C each. The space between their plates is now filled with dielectrics as shown. If all the three capacitors still have equal capacitances, obtain the relation between the dielectric constants `K, K_(1), K_(2), K_(3) and K_(4)`.

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New capacitance of `C_(1), C.=(Kepsilon_(0)A)/(d)" " …(i)`
Capacitance of `C_(2)`
`(1)/(C.)=(d)/(2epsilon_(0)K_(1)A)+(d)/(2epsilon_(0)K_(2)A)`
`(1)/(C.)=(d)/(2epsilon_(0)A) ((K_(1)+K_(2))/(K_(1)K_(2)))`
`C.=(epsilon_(0))/(d)((2K_(1)K_(2))/(K_(1)+K_(2)))" "...(ii)`
Capacitance of `C_(3)`
`C.=(epsilon_(0)K_(3)A)/(2d)+(epsilon_(0)K_(4)A)/(2d)`
`=(epsilon_(0)A)/(d)+((K_(3)+K_(4))/(2))" " ...(iii)`
From equation (i), (ii) and (iii)
`(Kepsilon_(0)A)/(d)=(epsilon_(0)A)/(d)((2K_(1)K_(2))/(K_(1)+K_(2)))`
`=(epsilon_(0)A)/(d)((K_(3)+K_(4))/(2))`
`rArr K=(2K_(1)K_(2))/(K_(1)+K_(2))=(K_(3)+K_(4))/(2)`
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