The charge entering the positive terminal of an element is given by the expression q(t)=-12e^{-2t} mC. The power delivered to the element is p(t)=2.4e^{-3t} W. Compute the current in the element, the voltage across the element, and the energy delivered to the element in the time interval 0 < t < 100 ms.
Solution:
To find the current, we need to find the derivative of the charge equation. Remember that:
i=\dfrac{d(q(t))}{dt}
i=\dfrac{d(-12e^{-2t})}{dt}
(take the derivative)
i=24e^{-2t} mA or i=0.024e^{-2t} A
(First equation gives us mA while the second gives us A)
To find the voltage, remember that:
v(t)=\dfrac{p(t)}{i(t)}
v(t)=\dfrac{2.4e^{-3t}}{0.024e^{-2t}}
v(t)=100e^{-t}
v(t)=\dfrac{2.4e^{-3t}}{0.024e^{-2t}}
v(t)=100e^{-t}
The energy delivered to the element can be found by:
\,\displaystyle W=\int p(t)\,dt
\,\displaystyle W=\int^{0.1}_{0} (2.4e^{-3t})\,dt
W=-0.8e^{-3t}\Big|^{0.1}_{0}
W=0.207 J
Final Answers:
i=0.024e^{-2t} A
v(t)=100e^{-t}
W=0.207 J
v(t)=100e^{-t}
W=0.207 J
W=−0.8e^(−3t) | 0->0.1 not 0.01 (100 ms to s = 100×10^-3=0.1 s
Thanks so much, we’ve fixed the error 🙂