The voltage across a 10-Ωresistor varies as V=(170 V) sin(100 πt) . (a) Is the current in the r

step 1 A register r equals to 10 om and it has a voltage v equals to 170 volt sine 100 pi t. Okay, this

The voltage across a 10-Ωresistor varies as V=(170 V)...

The voltage across a $10-\Omega$ resistor varies as $V=(170 \mathrm{~V}) \sin (100 \pi t) .$ (a) Is the current in the resistor (1) in phase with the voltage, (2) ahead of the voltage by $90^{\circ},$ or (3) lagging behind the voltage by $90^{\circ} ?$ (b) Write the expression for the current in the resistor as a function of time and determine the voltage frequency.

Key Concepts

Ohm's Law in AC Circuits

Ohm's Law, which relates voltage, current, and resistance by the relation V = IR, is a fundamental concept in AC circuit analysis. In a resistor, this law implies that the current and voltage are directly proportional. When the voltage is a sinusoidal function, the current obtained by dividing the voltage by the resistance retains the same frequency and phase as the voltage, with the amplitude scaled appropriately by the reciprocal of the resistance.

Phase Relationships in Resistive Circuits

In a purely resistive circuit, the voltage and the current vary in phase with each other. This means that their maximum values, zero crossings, and signs occur at the same instants of time. Unlike inductive or capacitive elements that introduce phase shifts of +90° or -90° respectively, resistors ensure a zero phase difference between voltage and current.

Sinusoidal Waveforms in AC Analysis

Sinusoidal waveforms are commonly used in AC circuit analysis because they provide a simple representation of alternating currents and voltages. Represented mathematically in the form V(t) = V? sin(?t), these functions characterize both amplitude and frequency. The properties of sine waves, such as periodicity and phase, are key in understanding the behavior of AC circuits.

Angular Frequency and Frequency Conversion

Angular frequency (?) is a measure of how rapidly a sine or cosine function oscillates, expressed in radians per second, and is related to the standard frequency (f) by the relation ? = 2?f. This conversion is essential for translating the parameters of a sinusoidal waveform from angular to cycles per second (Hz) in practical AC circuit analyses.

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