Question

. The charge entering the positive terminal of an element is ???? = 10 sin 4???????? m????, while the voltage across the element is ???? = 2 cos 4???????? ????. (i) Find the power delivered to the element at t = 0.3s. (ii) Calculate the energy delivered to the element between 0 and 0.6s.

Answer 1

P (t = 0.3) = 164.5 mW

W ( 0 < t < 0.6) = 78.34 mJ

Step-by-step explanation:

Given:

q (t) = 10*sin(4*pi*t) mC

V (t) = 2 *cos(4*pi*t) V

part a

The current i (t) flowing through the element is obtained as follows:

i (t) = dq / dt

= d (10*sin(4*pi*t)) / dt

= 40 * pi * cos (4*pi*t) mA

Next P(t) delivered to the element is obtained as follows:

P (t) = i (t)*V(t)

= 40 * pi * cos (4*pi*t) * 2 *cos(4*pi*t)

= 80*pi*(cos(4*pi*t))^2 mW

Finally the power delivered to element @ t = 0.3 s

P (t=0.3) = 80*pi*(cos(4*pi*0.3))^2 = 164.50 mW

Answer: P (t = 0.3) = 164.5 mW

part b

Energy delivered to the element time 0 to 0.6 s is obtained as follows:

`W (0 <t<0.6) = \int\limits {P(t)} \, dt\\\\ =\int {80*pi*(cos(4*pi*t))^2} \, dt\\\\= (5 sin (8*pi*t) + 40*pi*t )\limits^0.6_0 \\\\= 78.33715mJ`

Answer: W ( 0 < t < 0.6) = 78.34 mJ