Question

What is the power dissipated in AC circuit in which voltage and current given by
$$v=230\sin (\omega t+{\displaystyle \frac{\pi }{2}})$$ and $$I=20\sin \omega t$$ ?

Answer 1

Hint: Learn about the power dissipated through an AC circuit. The power dissipated from a circuit is given by the product of the voltage and current through the circuit. For an AC circuit the power dissipated is the average taken over a period of time

Formula used:
The average power dissipated from an AC circuit is given by,
$$P={\displaystyle \frac{1}{2}}{V}_0{I}_0\cos \phi$$
where, $$V_0$$ is the amplitude of the voltage applied $$I_0$$ is the amplitude of the current through the circuit $$\phi$$ is the phase difference between the current and the voltage.

Complete step by step answer:
We have given here the voltage of the circuit is $$v=230\sin (\omega t+{\displaystyle \frac{\pi }{2}})$$ where, amplitude of the voltage is $$V_0=230V$$ , $$\omega$$ is the frequency of the source applied $${\displaystyle \frac{\pi }{2}}$$ is the initial phase of the voltage. The current through the circuit is given by, $$I=20\sin \omega t$$ where, $$I_0=20A$$ is the amplitude of the current through the circuit$$\omega$$ is the frequency of the source applied. So, the phase difference between the current and voltage is $$\phi ={\displaystyle \frac{\pi }{2}}$$.

Now, we know that the average power dissipated from an AC circuit is given by,
$$P={\displaystyle \frac{1}{2}}{V}_0{I}_0\cos \phi$$
where, $$V_0$$ is the amplitude of the voltage applied $$I_0$$ is the amplitude of the current through the circuit $$\phi$$ is the phase difference between the current and the voltage.

So, here we have, amplitude of the voltage is $$V_0=230V$$ , amplitude of the current through the circuit $$I_0=20A$$and phase difference between the current and the voltage is $$\phi ={\displaystyle \frac{\pi }{2}}$$. Hence putting the values we have,
$$P={\displaystyle \frac{1}{2}}200\times 20\times \cos {\displaystyle \frac{\pi }{2}}$$
$$\Rightarrow P={\displaystyle \frac{1}{2}}200\times 20\times 0$$
$$\therefore P=0$$

Hence, the average power dissipated through the circuit is $$0watt$$.

Note: In the given circuit the maxima of voltage meets the minima of the current so the average value of the power becomes zero for a full circle. The average power dissipated through the circuit is zero does not mean that no energy is being exerted by the source in the circuit.