Question:
The instantaneous values of alternating current and voltages in a circuit are given as
$i=\frac{1}{\sqrt2}\sin(100\pi t)$ ampere
$e=\frac{1}{\sqrt2}\sin(100\pi t+\frac{\pi}{3})$ Volt .
The average power in watts consumed in the circuit is
The instantaneous values of alternating current and voltages in a circuit are given as
$i=\frac{1}{\sqrt2}\sin(100\pi t)$ ampere
$e=\frac{1}{\sqrt2}\sin(100\pi t+\frac{\pi}{3})$ Volt .
The average power in watts consumed in the circuit is
Updated On: Apr 28, 2025
- $\frac{1}{4}$
- $\frac{\sqrt3}{4}$
- $\frac{1}{2}$
- $\frac{1}{8}$
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The Correct Option is D
Solution and Explanation
Given : $i=\frac{1}{\sqrt2}\sin(100\pi t)$ ampere
Compare it with $i=i_0 \, \sin(\omega t)$ we get
$\hspace20mmi_0=\frac{1}{\sqrt2}A$
Given : $e=\frac{1}{\sqrt2}\sin\big(100\pi t+\frac{\pi}{3}\big)$ volt
Compare it with , we get
$\hspace20mme_0=\frac{1}{\sqrt2}V,\phi=\frac{\pi}{3}$
$\therefore \, \, i_{rms}=\frac{i_0}{\sqrt2}=\frac{\sqrt2}{\sqrt2}A=\frac{1}{2}A$
$\hspace20mm e_{rms}=\frac{e_0}{\sqrt2}=\frac{\frac{1}{\sqrt2}}{\sqrt2}V=\frac{1}{2}V$
Average power consumed in the circuit,
$\hspace20mmP=i_{rms} \, e_{rms} \, cos\phi$
$\hspace20mm=\big(\frac{1}{2}\big)\big(\frac{1}{2}\big)cos\frac{\pi}{3}=\big(\frac{1}{2}\big)\big(\frac{1}{2}\big)\big(\frac{1}{2}\big)=\frac{1}{8}W$
Compare it with $i=i_0 \, \sin(\omega t)$ we get
$\hspace20mmi_0=\frac{1}{\sqrt2}A$
Given : $e=\frac{1}{\sqrt2}\sin\big(100\pi t+\frac{\pi}{3}\big)$ volt
Compare it with , we get
$\hspace20mme_0=\frac{1}{\sqrt2}V,\phi=\frac{\pi}{3}$
$\therefore \, \, i_{rms}=\frac{i_0}{\sqrt2}=\frac{\sqrt2}{\sqrt2}A=\frac{1}{2}A$
$\hspace20mm e_{rms}=\frac{e_0}{\sqrt2}=\frac{\frac{1}{\sqrt2}}{\sqrt2}V=\frac{1}{2}V$
Average power consumed in the circuit,
$\hspace20mmP=i_{rms} \, e_{rms} \, cos\phi$
$\hspace20mm=\big(\frac{1}{2}\big)\big(\frac{1}{2}\big)cos\frac{\pi}{3}=\big(\frac{1}{2}\big)\big(\frac{1}{2}\big)\big(\frac{1}{2}\big)=\frac{1}{8}W$
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Concepts Used:
AC Voltage
When voltage changes its direction after every half cycle is known as alternating voltage. The current flows in the circuit at that time are known as alternating current. The alternating current(AC) follows the sine function which changes its polarity concerning time. Most of the electrical devices are operating on the ac voltage.
