In an a.c. circuit the voltage applied is E = E0 sin ?t. The resulting current in the circuit is I=Io sin { wt - p/2} The power consumption in the circuit is given by ?

In an alternating current (AC) circuit, the relationship between voltage and current can be quite fascinating, especially when we consider how they interact over time. The voltage applied is expressed as E = E0 sin(ωt), where E0 is the peak voltage, ω is the angular frequency, and t is time. The current, on the other hand, is given by I = Io sin(ωt - π/2), indicating that the current lags the voltage by 90 degrees (or π/2 radians). This phase difference is crucial for understanding power consumption in the circuit.

Understanding Power in AC Circuits

Power consumption in an AC circuit can be calculated using the formula:

• P = VI, where P is power, V is voltage, and I is current.

However, because both voltage and current are sinusoidal and can be out of phase, we need to consider the root mean square (RMS) values and the phase difference. The average power consumed in an AC circuit is given by:

• P_avg = V_rms * I_rms * cos(φ)

Here, φ represents the phase difference between the voltage and current. In your case, since the current lags the voltage by π/2, we have:

• φ = π/2

Calculating RMS Values

To find the RMS values of voltage and current, we use the following formulas:

• V_rms = E0 / √2
• I_rms = Io / √2

Substituting into the Power Formula

Now, substituting these RMS values into the average power formula:

P_avg = (E0 / √2) * (Io / √2) * cos(φ)

Since cos(π/2) = 0, we find:

P_avg = (E0 * Io / 2) * 0 = 0

Interpreting the Results

This result indicates that when the current lags the voltage by 90 degrees, the average power consumed in the circuit is zero. This situation typically occurs in purely inductive circuits, where energy is stored in the magnetic field and returned to the source, rather than being consumed as useful power.

Real-World Implications

In practical terms, this means that while the circuit may have a voltage and current flowing through it, no net power is being consumed over a complete cycle. This is an important concept in AC circuit analysis, especially when designing systems that involve inductors and capacitors, as they can lead to reactive power, which does not contribute to actual work done.

In summary, the power consumption in an AC circuit where the current lags the voltage by π/2 is zero, highlighting the unique characteristics of reactive components in electrical systems.