Question

In ac circuit with voltage \(\mathrm{V}\) and current \(\mathrm{I}\), the power dissipated is. (a) VI (b) \((1 / 2) \mathrm{VI}\) (c) \((1 / \sqrt{2}) \mathrm{VI}\) (d) Depends on the phase between \(\mathrm{V}\) and I

Short answer

The short answer is: (d) Depends on the phase between V and I, as the power in AC circuits is given by the formula \(P = VI \times \cos{\phi}\), where the power dissipated depends on the phase angle \(\phi\) between voltage and current.

Step-by-step solution

Power formula for AC circuits

The power in an AC circuit is given by the formula: \(P = VI \times \cos{\phi}\) Where P is the power dissipated, V is the voltage, I is the current, and \(\phi\) is the phase angle between the voltage and current waveforms.

Evaluate the given options

Now let's evaluate each option considering the power formula. (a) VI: This would be the power formula if there were no phase difference between voltage and current, i.e., \(\phi = 0\). However, in an AC circuit, the phase difference between voltage and current may not always be zero. (b) \((1 / 2) VI\): This is not related to the power formula in AC circuits, as it doesn't include the phase angle \(\phi\). (c) \((1 / \sqrt{2}) VI\): This is again not related to the power formula in AC circuits, as it doesn't include the phase angle \(\phi\). (d) Depends on the phase between V and I: This is the correct option, as the power in AC circuits is given by the formula \(P = VI \times \cos{\phi}\), where the power dissipated depends on the phase angle \(\phi\) between voltage and current. So, the correct option is (d) Depends on the phase between V and I.

Key concepts

Phase Angle

In an alternating current (AC) circuit, the phase angle \( \phi \) is an important concept that describes the offset between the voltage and the current waveforms. If you imagine the AC voltage and current as two waves that move up and down, the phase angle tells you how much one wave is shifted relative to the other.

This phase difference arises due to the reactive components like capacitors and inductors present in the circuit. These components affect how quickly the current can rise after the voltage starts to increase or decrease.

• When \( \phi = 0 \), the voltage and current are perfectly in sync, meaning they reach their peaks and zero values at the same time.
• If \( \phi \) is positive, the current lags behind the voltage.
• Conversely, if \( \phi \) is negative, the current leads the voltage.

Understanding the phase angle is essential because it directly influences how efficiently the circuit can perform work or dissipate power. It is important to measure this angle accurately to ensure optimal functioning of the AC system.

Power Formula

The power formula is a quintessential part of understanding AC circuits. In these circuits, the power dissipated is given by \[ P = VI \times \cos{\phi} \]where:

• \( P \) is the power dissipated, measured in watts (W).
• \( V \) is the voltage in volts (V).
• \( I \) is the current in amperes (A).
• \( \cos{\phi} \) is the cosine of the phase angle \( \phi \).

This formula shows that the real power (or active power) in an AC circuit depends not only on the voltage and current, but also significantly on the phase angle between them. The term \( \cos{\phi} \) is sometimes referred to as the power factor.
High power factor indicates more efficient power usage, while a low power factor can lead to increased power loss. This formula helps in determining how much actual work is done in the circuit, distinguishing from apparent power, which doesn't account for phase differences.

Voltage and Current

Voltage \( V \) and current \( I \) are two fundamental components of any electrical circuit, including AC circuits. They work together to transmit energy. Voltage is the potential difference between two points, and it essentially pushes electric charges through a circuit, creating current flow.

• Voltage is measured in volts (V) and can be thought of as the "pressure" pushing the current through the circuit.
• Current is measured in amperes (A) and is the rate at which charge flows through a point in the circuit.

In AC circuits, both voltage and current change direction and magnitude over time, following a sinusoidal waveform. The relationship between the two is not as straightforward as in DC circuits, due to the presence of the phase angle, which arises from the reactive elements in the circuit.

By analyzing both parameters and their phase relationship, one can design and operate AC systems effectively, ensuring that the devices connected to the circuit perform their intended functions properly with minimal energy loss.