## anonymous 5 years ago 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.

1. anonymous

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

2. anonymous

i=C dv/dt

3. anonymous

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

4. anonymous

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

5. anonymous

integrate both sides with respect to t

6. anonymous

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

7. anonymous

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