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Math Help - electrical circuit problem

  1. #1
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    electrical circuit problem

    Consider an electrical circuit comprising an inductor and a capacitor, with no external applied voltage, and no electrical resistance. The current, i, through the inductor is related to the charge, q, on the capacitor by the differential equation: where the inductance, L, and capacitance C, are positive constants.


    L(di/dt)+q/c=0


    (a)Given that the charge and the current are related through i=dq/dt derive a second order differential equation for q(t).


    i know that the second order differential is


    L(d2q/dt2)+q/c=0


    please help me solve the problem


    b)assuming that the initial charge q(0)=q0 and the initial current i(0)=i0, show that


    q(t)=q0cos(wt)+(i0/w)sin(wt)


    where w^2=1/LC

    c) hence show that q(t) can be written as

    q(t)=Acos(wt+fi)

    where A and fi are to be determined


    thanks for the help
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  2. #2
    GJA
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    Re: electrical circuit problem

    Hi, math254.

    Your DE is a second order linear homogeneous equation. One route you could take is the following:

    1. Set up what's called the "characteristic equation." This is done by replacing your DE with the polynomial

    L\lambda^{2}+\frac{1}{C}=0

    (if you haven't seen this before and need more details let me know).

    2. Solve the above for \lambda. Note: The two solutions for \lambda will be a complex numbers.

    3. The general solution to your DE is then

    q(t)=e^{\alpha t}(c_{1}\cos(\omega t)+c_{2}\sin(\omega t))

    where \alpha is the real part of \lambda and \omega is the imaginary part of \lambda. See Equation 11 of http://www.stewartcalculus.com/data/...arEqns_Stu.pdf

    4. Now use the initial conditions to determine c_{1} and c_{2}.

    5. To prove part c) I would suggest visiting http://www.mathcentre.ac.uk/resource...a-alphaetc.pdf. There is a nice outline showing how to express the sum of sine and cosine as one cosine function.

    Does this help sort out the confusion?

    Good luck!
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  3. #3
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    Re: electrical circuit problem

    thank you
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