In an AC circuit, the voltage applied is E = E_0 sin(wt). The resulting current in the circuit is I

Q: Why is the power zero if both current and voltage have non-zero amplitudes?

Power is the product of instantaneous voltage and current. Because they are $90^\circ$ out of phase, whenever voltage is at its maximum, current is zero, and vice versa. The positive power in one quarter-cycle is perfectly canceled by the negative power in the next, making the average zero.

Q: What does a phase difference of $-\pi/2$ physically mean?

It means the current reaches its peak value a quarter of a cycle after the voltage reaches its peak. This is characteristic of a purely inductive circuit, where the inductor's back-emf opposes changes in current.

Q: Is energy completely lost in this circuit?

No, energy is not lost at all! It simply oscillates. The source gives energy to the inductor to build a magnetic field, and then the inductor returns that exact energy back to the source. No energy is dissipated as heat.

Q: What is 'wattless current'?

Wattless current is the component of the AC current that is exactly $90^\circ$ out of phase with the voltage ($I_{rms} \sin \phi$). Because $\cos(90^\circ) = 0$, this specific component contributes zero watts to the average power dissipation.

Q: Would the answer change if the current equation had a $+\pi/2$ instead?

No, the average power would still be zero. A $+\pi/2$ phase shift means the circuit is purely capacitive. Since $\cos(\pi/2) = 0$, a pure capacitor also consumes zero average power.

Question

In an AC circuit, the voltage applied is $E = E_0 \sin \omega t$. The resulting current in the circuit is $I = I_0 \sin \left( \omega t - \frac{\pi}{2} ight)$. The power consumption in the circuit is given by

Options:

(A) $P = \frac{E_0 I_0}{\sqrt{2}}$

(B) $P = 	ext{zero}$

(C) $P = \frac{E_0 I_0}{2}$

(D) $P = \sqrt{2} E_0 I_0$

Sigma Percentile · Accepted Answer

### The Mystery of the Wattless Current

Imagine you are pushing a child on a swing. If you push exactly when the swing is moving forward, you transfer energy to it. But what if you push when the swing is at its highest point and momentarily stationary? You exert force, but you do no net work. This mechanical analogy perfectly captures the essence of what happens in certain Alternating Current (AC) circuits, specifically those involving pure inductors or capacitors.

In our problem, we are given the voltage applied to an AC circuit as $E = E_0 \sin \omega t$. The resulting current is given by $I = I_0 \sin \left( \omega t - \frac{\pi}{2} ight)$.

### Analyzing the Phase Difference

Look closely at the arguments of the sine functions. The voltage has a phase of $\omega t$, while the current has a phase of $\omega t - \frac{\pi}{2}$. This $-\frac{\pi}{2}$ is the critical piece of the puzzle. It tells us that the current is lagging behind the voltage by exactly $90^\circ$ (or $\frac{\pi}{2}$ radians).

In the world of AC circuits, a $90^\circ$ phase lag is the hallmark signature of a **purely inductive circuit**. The inductor opposes the change in current, causing the current wave to peak a quarter-cycle after the voltage wave peaks.

### The Master Equation for Power

To find the power consumed, we rely on the master formula for average power in an AC circuit:

$$P_{avg} = E_{rms} I_{rms} \cos \phi$$

Here, $E_{rms}$ and $I_{rms}$ are the root-mean-square values of voltage and current, and $\cos \phi$ is known as the **power factor**. The angle $\phi$ represents the phase difference between the voltage and the current.

### The Final Calculation

We have already established that the phase difference $\phi = \frac{\pi}{2}$. Let's substitute this into our power factor term:

$$\cos \phi = \cos\left(\frac{\pi}{2}ight) = 0$$

Because the power factor is exactly zero, the entire expression for average power collapses:

$$P_{avg} = E_{rms} I_{rms} 	imes 0 = 0$$

The circuit consumes absolutely **zero** net power over a complete cycle.

But wait, if voltage and current are both flowing, where does the energy go? The energy is simply playing a game of ping-pong! During one quarter of the cycle, the source supplies energy to build up the magnetic field inside the inductor. In the next quarter, the collapsing magnetic field returns that exact same energy back to the source. Because no energy is dissipated as heat (unlike in a resistor), the net power consumption is zero. The current that flows in such a circuit is beautifully termed as **wattless current**.

Animated Solution for Physics - Alternating Current: In an AC circuit, the voltage applied is $E = E_{0} sin ω t$ . The resulting current in the circuit is $I = I_{0} sin (ω t - \frac{π}{2})$ . The power consumption in the circuit is given by

Select Answer:

Visualized Solution

Analyzing $E$ and $I$ Equations

Average Power Formula

Identifying Phase Difference $ϕ$

Calculating Power Factor

Final Average Power

The Way Forward

The Sigma Insight: AC Circuits and Power in AC Circuits

The Mystery of the Wattless Current

Analyzing the Phase Difference

The Master Equation for Power

The Final Calculation

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