Problem 2 - Applied Physics

Question

For an ac circuit with \(R=375 \Omega, C=5.00 \mu \mathrm{F}, E=20.0 \mathrm{~V}\), and \(f=1.00 \mathrm{kHz}\) : (a) find the impedance (in ohms), (b) find the phase angle, and (c) find the current.

Step-by-Step Solution

Answer

(a) Impedance: 376 Ω, (b) Phase angle: -4.84°, (c) Current: 0.053 A.

Step 1: Calculate the Capacitive Reactance

To find the capacitive reactance \( X_C \), use the formula:\[ X_C = \frac{1}{2\pi f C} \] where \( f \) is the frequency and \( C \) is the capacitance. Substitute \( f = 1000 \) Hz and \( C = 5.00 \times 10^{-6} \) F into the formula:\[ X_C = \frac{1}{2\pi (1000)(5.00 \times 10^{-6})} \approx 31.8 \Omega \]

Step 2: Calculate the Impedance

The impedance \( Z \) in an RC circuit can be calculated using the formula:\[ Z = \sqrt{R^2 + X_C^2} \]Substitute \( R = 375 \) \( \Omega \) and \( X_C = 31.8 \) \( \Omega \):\[ Z = \sqrt{(375)^2 + (31.8)^2} \approx 376 \Omega \]

Step 3: Calculate the Phase Angle

The phase angle \( \phi \) (in degrees) is found using the formula:\[ \phi = \arctan\left( \frac{-X_C}{R} \right) \]Using \( R = 375 \) \( \Omega \) and \( X_C = 31.8 \) \( \Omega \):\[ \phi = \arctan\left( \frac{-31.8}{375} \right) \approx -4.84^\circ \]

Step 4: Calculate the Current

The current \( I \) in the circuit is given by Ohm's law for AC circuits:\[ I = \frac{E}{Z} \]Substitute \( E = 20.0 \) V and \( Z = 376 \) \( \Omega \):\[ I = \frac{20.0}{376} \approx 0.053 \text{ A} \]

Key Concepts

Impedance Calculation

When working with AC circuits, understanding impedance is crucial. Impedance is the total resistance a circuit offers to the flow of alternating current. It is represented by the symbol \( Z \) and is a complex quantity, having both resistive and reactive components. In an RC circuit, impedance can be determined using:

• Resistance \( R \)
• Capacitive reactance \( X_C \)

The formula to calculate impedance in an RC circuit is:\[ Z = \sqrt{R^2 + X_C^2} \]This suggests that the impedance is the vector sum of the resistance and the capacitive reactance. It’s essentially the hypotenuse of a right triangle formed by the resistance and capacitive reactance when plotted on a complex plane. Calculating impedance helps in understanding how much current the circuit allows at a given voltage and frequency.
It also provides insight into how the circuit opposes current flow, not just through resistance, but also through any phase shift caused by capacitive influences.

Capacitive Reactance

Capacitive reactance is a measure of a capacitor's opposition to alternating current. It is frequency-dependent and plays an important role in AC circuits with capacitors. The formula for capacitive reactance \( X_C \) is:\[ X_C = \frac{1}{2\pi f C} \]

• \( f \) is the frequency of the AC signal
• \( C \) is the capacitance

Capacitive reactance decreases with higher frequency. This means that at higher frequencies, capacitors allow more current to pass through because their reactance decreases. At lower frequencies, the capacitive reactance is higher, meaning that the capacitor blocks more current. Understanding this relationship is crucial for analyzing how capacitors affect the behavior of AC circuits. Knowing the capacitive reactance in an AC circuit aids in determining the overall impedance and phase angle, which are important for optimizing circuit performance.

Phase Angle

In AC circuits, the phase angle \( \phi \) represents the shift between the voltage and the current waveforms. This phase difference is influenced by the capacitive and resistive components of the circuit. It is calculated using the formula:\[ \phi = \arctan\left( \frac{-X_C}{R} \right) \]

• Negative values indicate that the current lags the voltage, typical in RC circuits
• The phase angle is usually measured in degrees

The phase angle reflects how much the voltage leads or lags the current, which can impact the timing of power delivery in AC circuits. Understanding phase angles is essential for applications such as power factor correction and efficient energy use. Engineers often use phase angles to assess the efficiency and stability of circuit operations. A small phase angle means minimal phase difference, indicating more efficient energy transfer. Grasping this concept is vital for students who wish to excel in AC circuit analysis.

Ohm's Law for AC Circuits

Ohm's Law is fundamental in understanding electrical circuits, and it applies to AC circuits as well, albeit with some modifications. Unlike DC circuits, where relationships are purely linear, AC circuits involve impedance. The modified version of Ohm's Law for AC circuits can be expressed as:\[ I = \frac{E}{Z} \]

• \( I \) is the current
• \( E \) is the applied voltage
• \( Z \) is the impedance

This equation tells us that the current in an AC circuit is inversely proportional to the impedance, just like in the DC form of Ohm's Law. Understanding this concept allows for the analysis of how AC components like resistors and capacitors interact together to influence circuit behavior. This foundational knowledge is critical for anyone looking to delve deeper into the analysis and design of AC circuits. By grasping Ohm's Law for AC circuits, students can predict how changes in voltage or circuit elements will affect current flow.