anonymous
  • anonymous
The current through a 4F capacitor is given by the equation i(t)=5e^(-t/2) for t>0. Assuming that the capacitor is is initially uncharged, determine an expression for the voltage across the capacitor.
Mathematics
jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
i=dq/dt dq=i(dt) integrate to get the charge use charge=capacitance*voltage
anonymous
  • anonymous
i=C dv/dt
anonymous
  • anonymous
therefore 4 dv/dt = 5e^(-t/2)

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anonymous
  • anonymous
so \[\frac{dv}{dt} = \frac{5}{4}e^{-\frac{t}{2}}\]
anonymous
  • anonymous
integrate both sides with respect to t
anonymous
  • anonymous
\[v = -\frac{5}{2}e^{-\frac{t}{2}} +C \]
anonymous
  • anonymous
now, it is initially uncharged, so when t=0, v=0 that gives the constant C = 5/2 \[v(t) = \frac{5}{2} ( 1 - e^{-\frac{t}{2}}) V \]

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