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Question
Find the ratio of the potential differences that must be applied across the parallel and series combination of two capacitors C1 and C2 with their capacitances in the ratio 1 : 2 so that the energy stored in the two cases becomes the same.
Solution
Data:`C_1/C_2 = 1/2,U_1("for parallel") = U_2("for series")`
`"C"_1/"C"_2 = 1/2`
∴ C2 = 2C1
For the parallel combination of C1 and C2,
Cp = C1 + C2 = 3C1
and charged to a potential V1, the energy stored is
`"u"_1 = 1/2"C"_"p""V"_1^2 = 3/2 "C"_1"V"_1^2`
For the series combination of C1 and C2,
`"C"_"s" = ("C"_1"C"_2)/("C"_1 + "C"_2) = (2"C"_1^2)/(3"C"_1) = 2/3 "C"_1`
and charged to a potential V2, the energy stored is
`"u"_2 = 1/2"C"_"s""V"_2^2 = 1/3 "C"_1"V"_2^2`
∴ For `"u"_1 = "u"_2, 3/2"C"_1"V"_1^2 = 1/3"C"_1"V"_2^2`
∴ `("V"_1/"V"_2)^2 = 2/9`
∴ `"V"_1/"V"_2 = sqrt2/3 = 1.414/3 = 0.471`
This gives the required ratio.
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