AC Voltage Applied to a Resistor: Class 12 Physics Explained

Introduction:

The study of alternating current (AC) circuits forms a significant part of Class 12 Physics. When an AC voltage is applied to a resistor, it produces a sinusoidal current that remains in phase with the voltage. This topic is crucial for understanding electrical circuits and their applications. In this guide, we will explore the behavior of AC voltage in resistive circuits, supported by mathematical analysis, diagrams, and practical examples.

What is AC Voltage Applied to a Resistor?

Alternating current (AC) continuously changes its magnitude and direction. A resistor, a purely resistive element, opposes the flow of electric current based on Ohm's Law. When AC voltage is applied across a resistor, the relationship between current and voltage remains linear.

The Mathematical Representation:

AC Voltage: V(t) = V_0 sin(ωt)

Current Through Resistor: I(t) = I_0 sin(ωt)

- V_0: Peak voltage

- I_0: Peak current

- ω: Angular frequency (ω = 2πf)

- f: Frequency of AC source

According to Ohm's Law:

I_0 = V_0 / R

Where R is the resistance in ohms (Ω).

Voltage-Current Relationship:

Voltage and current are in phase in a resistive AC circuit. This means the peaks of voltage and current occur at the same time. No phase difference (φ = 0°).

Power Dissipation in Resistor:

Power dissipated in a resistor is given by:

P(t) = V(t) · I(t)

- Instantaneous Power: P(t) = V_0 I_0 sin²(ωt)

- Average Power: P_avg = 1/2 V_0 I_0 = I_rms² R = V_rms² / R

- V_rms and I_rms are the root mean square values:

V_rms = V_0 / √2, I_rms = I_0 / √2

Phasor Diagram for a Resistive Circuit:

A phasor is a graphical representation of sinusoidal quantities. In a purely resistive circuit:

- The voltage and current phasors align, indicating they are in phase. The diagram illustrates the angular frequency ωt, and both vectors rotate synchronously.

Key Observations:

1. In an AC circuit with a resistor:

- Voltage and current are directly proportional.

2. The circuit does not store energy but dissipates it as heat.

3. The power factor (cos φ) equals 1, indicating maximum power efficiency.

Conclusion:

The study of AC voltage applied to a resistor highlights the foundational principles of AC circuits. The linear relationship between voltage and current and the absence of phase difference make resistive circuits simple yet critical for practical applications in electrical engineering. This understanding serves as a stepping stone to analyzing more complex circuits involving inductors and capacitors.