Question

Calculate the average current during the first quarter-cycle.

• A. $0.4A$
• B. $0.8A$
• C. $0.2A$
• D. $0.9A$

Answer 1

Answer: (A)

$0.4A$

Given : $C=1\mu F$
Voltage $V=100$ volts
Frequency of oscillation $f={10}^{3}Hz$
Time period $T=\frac{1}{f}={10}^{-3}{s}^{-1}$
Angular frequency $w=2\pi f=2\pi \times {10}^3=6.28\times {10}^{3}rad/s$
Charge as a function of time is given by $q=CV\cos wt$
$\therefore$ $q={10}^{-6}\times 100\cos (6.28\times {10}^{3}t)={10}^{-4}\cos (6.28\times {10}^{3}t)$
Current flowing $i=|\frac{dq}{dt}|=CVw\sin wt$
$⟹i={10}^{-6}\times 100\times 6.28\times {10}^3\sin wt=0.628\sin wt$
Average current during first quarter cycle $i_{avg}=\frac{{\int }_0^{T/4}idt}{T/4}$
$⟹$ cycle $i_{avg}=\frac{{\int }_0^{T/4}0.628\sin wtdt}{T/4}$
Or $i_{avg}=2.5\times {10}^3\times \frac{\cos wt}{w}{|}_0^{T/4}$
Or $i_{avg}=2.5\times {10}^3\times \frac{1}{6.28\times {10}^3}=0.4A$