Answer

Verified

87.3k+ views

**Hint:**In an alternating current circuit, values of both electromotive force and current change continuously with respect to time. So, we cannot calculate the power directly. the average power for the sinusoidal function will be obtained by averaging over one period. The average power of an alternating current circuit is also known as the true power of the electrical circuit.

First we calculate root mean square (rms) values of current and voltage. Then we find the phase difference between them. Phase difference is denoted by symbol. In question phase difference is \[\dfrac{\pi }{3}\]in radian or\[{60^0}\].

**Formula used:**We know that the power is equal to the product of voltage and time. We calculate average power by the formula \[{P_{av}} = {V_{rms}} \times {i_{rms}}\cos \phi \].

**Complete step by step solution:**

Given: \[i = \dfrac{1}{{\sqrt 2 }}\sin \left( {100\pi t} \right)Ampere\] and \[e = \dfrac{1}{{\sqrt 2 }}\sin \left( {100\pi t + \dfrac{\pi }{3}} \right)Volt\]

i.e., \[{i_0} = \dfrac{1}{{\sqrt 2 }}\] and \[{V_0} = \dfrac{1}{{\sqrt 2 }}\].

Therefore, average power consumed in the circuit is given by

\[{P_{av}} = {V_{rms}} \times {i_{rms}}\cos \phi \].

\[{P_{av}} = \dfrac{1}{2} \times \dfrac{1}{2} \times \cos {60^0}\]

We know that \[{V_{rms}} = \dfrac{1}{2}\], \[{i_{rms}} = \dfrac{1}{2}\]and \[\phi = {60^0}\]

\[ \Rightarrow {P_{av}} = \dfrac{1}{2} \times \dfrac{1}{2} \times \dfrac{1}{2} = \dfrac{1}{8}W\]

\[\therefore {P_{av}} = \dfrac{1}{8}W\]

**Hence, Average power consumed in AC circuit is \[\therefore {P_{av}} = \dfrac{1}{8}W\].**

**Additional information:**For AC circuits, the instantaneous power is constantly changing with varying amounts of time. The curve of power vs time is sinusoidal. As we get this type of curve i.e., it makes it difficult to measure power. Calculating average or mean value of the power, It is therefore more convenient to calculate. The average value of the instantaneous power over a fixed number of cycles, the average power of sinusoidal curve is given simply as: \[{P_{av}} = {V_{rms}} \times {i_{rms}}\cos \phi \].

**Note:**Students must be careful to calculate root mean square values of current and voltage. They must be careful to find phase difference between voltage and current. In question rms values are given. Don’t put \[\dfrac{1}{{\sqrt 2 }}\]as \[{V_{rms}}\] and \[{i_{rms}}\]in formula. Here \[\dfrac{1}{{\sqrt 2 }}\]is \[{i_0}\] for current and \[\dfrac{1}{{\sqrt 2 }}\]is \[{V_0}\].

Recently Updated Pages

Name the scale on which the destructive energy of an class 11 physics JEE_Main

Write an article on the need and importance of sports class 10 english JEE_Main

Choose the exact meaning of the given idiomphrase The class 9 english JEE_Main

Choose the one which best expresses the meaning of class 9 english JEE_Main

What does a hydrometer consist of A A cylindrical stem class 9 physics JEE_Main

A motorcyclist of mass m is to negotiate a curve of class 9 physics JEE_Main

Other Pages

If a wire of resistance R is stretched to double of class 12 physics JEE_Main

A lens forms a sharp image on a screen On inserting class 12 physics JEE_MAIN

Which of the following facts regarding bond order is class 11 chemistry JEE_Main

Velocity of car at t 0 is u moves with a constant acceleration class 11 physics JEE_Main

Find the number of nitrates which gives NO2gas on heating class 12 chemistry JEE_Main

If temperature of sun is decreased by 1 then the value class 11 physics JEE_Main