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La tensión en el capacitor se obtiene integrando la corriente y dividiendo entre la capacitancia, es decir:
v(t)=1C∫0ti(τ)dτv(t)=\frac{1}{C}\int_0^t i(\tau)d\tau
Con ( C=0.5 , \text{F} ) y
i(t)=6(1−e−t) A,i(t)=6\bigl(1-e^{-t}\bigr) \, \text{A},
se tiene:
v(t)=10.5∫0t6(1−e−τ)dτ=2⋅6∫0t(1−e−τ)dτ=12[∫0t(1−e−τ)dτ].v(t)=\frac{1}{0.5}\int_0^t 6\bigl(1-e^{-\tau}\bigr)d\tau = 2\cdot6\int_0^t (1-e^{-\tau})d\tau = 12\left[\int_0^t\left(1-e^{-\tau}\right)d\tau\right].
Se evalúa la integral:
∫0t(1−e−τ)dτ=[t+e−t−1],\int_0^t \left(1-e^{-\tau}\right)d\tau = \left[t+e^{-t}-1\right],
por lo que:
v(t)=12[t+e−t−1].v(t)=12\bigl[t+e^{-t}-1\bigr].
En ( t=2 , \text{s} ):
v(2)=12[2+e−2−1]=12[1+e−2].v(2)=12\bigl[2+e^{-2}-1\bigr] = 12\bigl[1+e^{-2}\bigr].
Utilizando ( e^{-2}\approx0.1353 ):
v(2)≈12(1+0.1353)≈12×1.1353≈13.62 V.v(2)\approx 12\bigl(1+0.1353\bigr) \approx 12\times1.1353\approx 13.62\, \text{V}.
La tensión en el capacitor a ( t=2 , \text{s} ) es aproximadamente 13.62 V.
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