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Question
Capacitors P and Q have identical cross-sectional areas A and separation d. The space between the capacitors is filled with a dielectric of dielectric constant Er as shown in the figure. Calculate the capacitance of capacitors P and Q.
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Solution
- The arrangement can be supposed to be a parallel combination of two capacitors each with plate area A/2 and separation. d. Total capacitance `"C"_"p" = "C"_"air" + "C"_"dielectric"`
`= (epsilon_0("A"//2))/"d" + (epsilon_0("A"//2)epsilon"r")/"d"`
`"C"_"p" = (epsilon_0"A")/"2d" (1 + epsilon_"r")` - The arrangement can be supposed to be a series combination of two capacitors, each with plate area A and separation d/2.
Total capacitance CQ = `("C"_1"C"_2)/("C"_1 + "C"_2)`
`= ((2epsilon_0"A")/"d" xx (2epsilon_"r"epsilon_0"A")/"d")/((2epsilon_0"A")/"d" + (2epsilon_"r"epsilon_0"A")/"d")`
`= (4epsilon_"r" ((epsilon_0"A")/"d")^2)/((2epsilon_0"A")/"d" (1 + epsilon_"r"))`
`= (2epsilon_"r" (epsilon_0"A")/"d")/(1 + epsilon_"r")`
`"C"_"Q" = (2epsilon_0"A")/"d" (epsilon_"r"/(1 + epsilon_"r"))`
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