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21 December, 17:18

When two capacitors are connected in parallel and then connected to a battery, the total stored energy is 6.7 times greater than when they are connected in series and then connected to the same battery. What is the ratio of the two capacitances?

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  1. 21 December, 17:26
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    4.48 and 0.22

    Explanation:

    Below are the stated formulas for capacitors in series and parallel.

    Let both capacitors be C1 and C2 respectively.

    For parallel:

    Total C = C1 + C2

    For series (after rearranging the formular):

    Total C = C1C2 / (C1 + C2)

    Now, formular to calculate total energy stored is given below.

    Parallel:

    E = 1/2 * CV^2

    = 1/2 * (C1 + C2) * V^2

    Series:

    E = 1/2 * CV^2

    = 1/2 * (C1C2 / (C1 + C2)) * V^2

    In the question, it states that energy stored in parallel is 6.7 times the energy when stored in series.

    E (parallel) = 6.7E (series)

    1/2 * (C1 + C2) * V^2 = 6.7 * 1/2 * (C1C2 / (C1 + C2)) * V^2

    (C1 + C2) = 6.7C1C2 / (C1 + C2) (cancelling out V^2 and 1/2 on both sides)

    (C1 + C2) ^2 = 6.7C1C2

    C1^2 + 2C1C2 + C2^2 = 6.7C1C2

    C1^2 - 4.7C1C2 + C2^2 = 0

    Now, we need to divide through by C2^2 to get an equation of the form ax^2 + bx + c = 0, which we can then solve using the quadratic equation.

    C1^2 / C2^2 - 4.7C1 / C2 + 1 = 0

    let x = C1 / C2 (x is the ratio of the capacitance of the capacitors)

    x^2 - 4 x + 1 = 0

    Using the quadratic equation, the solutions are:

    x = C1 / C2 = 4.48 and x = C1 / C2 = 0.22 in two decimal places.
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