The current through a 4F capacitor is given by the equation i(t)=5e^(-t/2) for t0. Assuming that the capacitor is is initially uncharged, determine an expression for the voltage across the capacitor.

Question

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.

anonymous · Answer

i=dq/dt
dq=i(dt)
integrate to get the charge
use charge=capacitance*voltage

anonymous · Answer

i=C dv/dt

anonymous · Answer

therefore 4 dv/dt = 5e^(-t/2)

anonymous · Answer

so $$\frac{dv}{dt} = \frac{5}{4}e^{-\frac{t}{2}}$$

anonymous · Answer

integrate both sides with respect to t

anonymous · Answer

$$v = -\frac{5}{2}e^{-\frac{t}{2}} +C $$

anonymous · Answer

now, it is initially uncharged, so when t=0, v=0

that gives the constant C = 5/2

$$v(t) = \frac{5}{2} ( 1 - e^{-\frac{t}{2}}) V $$