91Ó°ÊÓ

Angular frequency is a critical concept when working with alternating current (AC) circuits, such as those involving capacitors and resistors in series, like the one in the original exercise. It is a measure of how quickly an AC voltage waveform oscillates in radians per second. The formula for angular frequency is given by \( \omega = 2\pi f \), where

• \( \omega \) represents the angular frequency in radians per second.
• \( f \) is the frequency of the waveform in hertz (Hz).

For our problem, we were provided directly with \( \omega = 120 \, \text{rad/s} \). Angular frequency helps to determine how the capacitive reactance, \( X_C \), can vary with the frequency of the applied voltage.
You can think of it as a rotational equivalent of the regular frequency, crucial for deriving other AC circuit parameters.

A series circuit is one where electrical components are connected end-to-end, so the same current flows through each component in succession. In the exercise, the resistor and capacitor are connected in series:

• The current flowing through each element is the same.
• The total voltage across the series circuit is the sum of voltages across each element.
• This circuit is subject to Ohm's Law, which we'll discuss in detail.

For a series RC circuit like this one, while the current remains constant, the capacitive reactance dictates how the voltage is shared between the capacitor and resistor.
This setup is particularly interesting because it combines resistive effects with reactive effects from capacitors, allowing the analysis of phase shifts between voltage and current.

Ohm's Law is foundational in electrical engineering, particularly in analyzing circuits. It states that the voltage (\( V \)) across a resistor is directly proportional to the current (\( I \)) flowing through it, expressed with:\[ V = I \times R \]where

• \( V \) is the voltage across the resistor in volts (V).
• \( I \) is the current in amperes (A).
• \( R \) is the resistance in ohms (\( \Omega \)).

In the step-by-step solution, Ohm's Law was used to find the voltage across the resistor by multiplying the current by the resistance: \( v_R(t) = 1.10 \cos(120t) \, \text{V} \). This principle helps analyze how AC signals interact with resistors, establishing the basis for calculating voltage across other components like our capacitor in a circuit.

Differential calculus is an essential mathematical tool used to find rates of change, and it's crucial in analyzing alternating signals in physics and engineering. In the context of a capacitive circuit, differential calculus helps us determine the behavior of current when we know how voltage changes over time.
In the exercise, we differentiated the voltage equation to find the expression for current: \( i(t) = C \frac{d}{dt} v_C \).

• This represents the proportionality between current and the rate of change of voltage across a capacitor.
• The derivative of the sinusoidal voltage function, \( v_C(t) = 7.60 \sin(120t) \), led to the cosine function expressing current: \( 4.38 \times 10^{-3} \cos(120t) \, \text{A} \).

Differential calculus thus gives us insight into the dynamic behaviors of circuits over time, crucial in circuits involving capacitors and alternating currents.