An alternating emf source with a variable frequency fd is...

An alternating emf source with a variable frequency $f_{d}$ is connected in series with a $50.0 \Omega$ resistor and a $28.0 \mu \mathrm{F}$ capacitor. The emf amplitude is $12.0 \mathrm{~V}_{-}$(a) Draw a phasor diagram for phasor $V_{R}$ (the potential across the resistor) and phaser $V_{C}$ (the potential across the capacitor). (b) At what driving frequency $f_{d}$ do the two phasors have the same length? At that driving frequency, what are (c) the phase angle in degrees, (d) the angular speed at which the phasors rotate, and (e) the current amplitude?

Key Concepts

Phasor Diagrams

Phasor diagrams represent sinusoidally varying quantities as rotating vectors in the complex plane, where the length of each vector indicates the amplitude and its angle represents the phase. These diagrams offer a visual method to analyze AC circuits by showing the phase relationships between different voltages or currents, thereby simplifying the calculation of resultant quantities.

Complex Impedance

Complex impedance combines resistance and reactance into a single complex quantity that characterizes how circuit elements oppose AC current. In AC circuit analysis, resistors contribute to the real part while capacitors (and inductors) contribute to the imaginary part, enabling the use of phasor methods to compute current, voltage drops, and phase differences.

RC Circuit Analysis

RC circuit analysis involves studying the behavior of circuits containing resistors and capacitors when subjected to alternating current. In these circuits, the resistor voltage is in phase with the current while the capacitor voltage lags by 90 degrees, a difference that can be effectively analyzed using phasor diagrams and complex impedance methods.

Phase Angle

The phase angle in AC circuits quantifies the difference in phase between voltage and current, or between voltages across different circuit elements. It is critical for understanding how the reactive components (like capacitors) influence the overall behavior of a circuit, affecting both the timing and magnitude of the observed signals.

Angular Frequency

Angular frequency, defined as ? = 2?f, where f is the driving frequency, provides a measure of how quickly the phasors (or the sinusoidal quantities) complete a cycle in radians per second. It plays a central role in relating time-dependent behavior to the phasor representations used in AC circuit analysis.

AC Current and Voltage Relationships

In AC circuits, the amplitude of current and voltage across different elements is determined by applying Ohm’s law in its phasor form, which takes into account both the magnitude and phase of the complex impedance. This relationship is essential for analyzing how different circuit configurations influence the overall current distribution and voltage drops.

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