Question

The charge entering the positive terminal of an element is q = 5 sin 4 π t mC while the voltage across the element (plus to minus) is v = 3 cos 4 π t V
(a) Find the power delivered to the element at t = 0.3 s.
(b) Calculate the energy delivered to the element between 0 and 0.6 s.

Answer 1

Answer:

a)123.37 mW

b) 58.75 mJ

Explanation:

a) The current (i) flowing through the terminal is given as:

$i=\frac{dq}{dt} \\i=\frac{d(5sin(4\pi t)}{dt} =20\pi cos(4\pi t)\ mA$

The Power delivered to the element (P(t)) is given as:

$P(t)=v(t)i(t)= 3 cos(4\pi t)*20\pi cos(4\pi t)*10^{-3}\\P(t)=60\pi cos^2(4\pi t)*10^{-3}\\at \ t \ =\ 0.3s,\ the \ power\ delivered\ is:\\P(0.3) = 60\pi cos^2(4\pi *0.3)*10^{-3}=123.37mW$

b)  the energy delivered to the element (W) between 0 and 0.6 s is given as:

$W = \int\limits^0_{0.6} {P(t)} \, dt \\W=\int\limits^0_{0.6} {60\pi cos^{2}(4\pi t)} \, dt =58.75mJ$