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Charging and discharging capacitor, differential equation

  1. Oct 19, 2015 #1
    1. The problem statement, all variables and given/known data
    A circuit consists of a voltage source, voltage ##V## , a resistor, resistance ##R##, and a capacitor, capacitance ##C##, in series.

    (i) Show that the charge ##Q(t)## in the capacitor satisfies the equation ##R Q' (t) + Q(t)/C = V ##.

    (ii) Suppose that ##R##, ##C## and ##V## are constant and that ##Q## is initially zero. Find an expression for ##Q(t)##. Sketch the solution.

    (iii) Suppose instead that ##V = V_{0}## for ##t < T## and ##V = 0## for ##t > T##, where ##V_{0}## is constant and ##Q## is again initially zero. Find the new expression for ##Q(t)##, both for ##t < T## and ##t > T##.

    2. Relevant equations


    3. The attempt at a solution

    I have managed to answer part 1 and 2. I used kirchoff's loop rule to show that the charge in the capacitor satisfies the given equation, and solved it to find ##Q(t)##.

    I found that ##Q(t) = vc(1 - e^{\frac{-t}{RC}})## which is a pretty standard result and satisfies the condition that ##Q(0) = 0##.

    I am stuck with part 3. I know that discharging capacitors obey ##Q = Q_{T} e^{-\frac{t}{RC}}##, where ##Q_{T}## is the charge on the capacitor when the voltage is set to zero. (By solving the same differential equation with V = 0)

    I have tried writing the charge on the capacitor as a piecewise function for t < T and t > T, but each expression does not equal the other at t = T.

    Am I supposed to find a single expression that encapsulates this behaviour?
     
  2. jcsd
  3. Oct 19, 2015 #2

    NascentOxygen

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    Staff: Mentor

    Vo adds charge as a decaying exponential to the capacitor throughout time interval T. Then, starting with this value of charge, it drains away in a falling exponential.
     
  4. Oct 19, 2015 #3
    ##Q_{t < T} = CV_{0}(1 - e^{-\frac{t}{RC}})##

    ##Q_{t=T} = CV_{0}(1 - e^{-\frac{T}{RC}})##

    ##Q_{t > T} = Q_{t=T} e^{-\frac{t}{RC}}##

    Like this?
     
  5. Oct 19, 2015 #4

    NascentOxygen

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    Staff: Mentor

    That looks right.
     
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