Problem 4 - Applied Physics

Question

Find the resonant frequency in each ac circuit. \(L=2.65 \mu \mathrm{H}\) and \(C=35.0 \mu \mathrm{F}\)

Step-by-Step Solution

Answer

The resonant frequency is approximately 16,519 Hz.

Step 1: Understand Resonant Frequency Formula

The resonant frequency for an LC circuit is determined using the formula \( f_r = \frac{1}{2\pi\sqrt{LC}} \), where \(L\) is the inductance and \(C\) is the capacitance.

Step 2: Convert Units

Ensure the units for inductance \(L\) and capacitance \(C\) are in Henry (H) and Farad (F) respectively. Here, \(L=2.65 \times 10^{-6} \text{ H}\) (since 1 \( \mu \text{H} = 10^{-6} \text{ H} \)) and \(C=35.0 \times 10^{-6} \text{ F}\) (since 1 \(\mu \text{F} = 10^{-6} \text{ F} \)).

Step 3: Substitute Values

Substitute \(L\) and \(C\) into the formula: \( f_r = \frac{1}{2\pi\sqrt{2.65 \times 10^{-6} \times 35.0 \times 10^{-6}}} \).

Step 4: Calculate the Expression Under the Square Root

First, calculate \( LC = 2.65 \times 10^{-6} \times 35.0 \times 10^{-6} = 9.275 \times 10^{-11} \).

Step 5: Find the Square Root

Compute the square root of \( 9.275 \times 10^{-11} \), which is approximately \( 9.63 \times 10^{-6} \).

Step 6: Compute Resonant Frequency

Substitute the square root back into the equation to find \( f_r = \frac{1}{2\pi \times 9.63 \times 10^{-6}} \).

Step 7: Final Calculation

Perform the final calculation to find \( f_r \), which gives approximately \( 16,519 \text{ Hz} \).

Key Concepts

LC Circuit

An LC circuit, also known as a resonant circuit, is composed of an inductor (L) and a capacitor (C) connected together in a loop. These components store energy in the form of a magnetic field and an electric field respectively. When connected to an AC source, the circuit can oscillate between storing energy in the inductor and capacitor.
This oscillation happens at a particular frequency known as the resonant frequency, denoted as \( f_r \). At this frequency, the impedance of the circuit is minimized, and the current reaches its maximum potential.

• The resonant condition occurs where the inductive reactance equals the capacitive reactance.
• The circuit behaves differently in at very high or low frequencies.

This unique property makes LC circuits quite useful in applications like tuning radio frequencies and signal filtering.

Inductance and Capacitance

Inductance and capacitance are fundamental electrical properties describing how circuits store and release energy. Inductance, measured in henries (H), relates to the magnetic field generated around a coil of wire when current flows through it. It characterizes how effectively a circuit can induce voltage through shifting magnetic fields.
Conversely, capacitance, measured in farads (F), is the ability of a system to store an electric charge. Capacitors hold energy in an electric field, which can quickly be released into a circuit.

• A higher capacitance allows more energy storage at a lower voltage.
• High inductance means stronger resistance to changes in current due to magnetic field effects.

These two properties work harmoniously in an LC circuit to determine the resonant frequency as shown in the formula \( f_r = \frac{1}{2\pi\sqrt{LC}} \). It emphasizes the significance of inductance and capacitance in predicting how an LC circuit will behave.

Physics Education

Physics education plays a critical role in understanding the principles behind electronic components like LC circuits. Building a foundational understanding of concepts like resonance, inductance, and capacitance is crucial for students looking to master AC circuit analysis.
Hands-on experiments and problem-solving exercises enhance this learning process, allowing for the practical application of theoretical knowledge.

• Experimenting with LC circuits in labs helps solidify theoretical concepts.
• Using simulations can visualize how these components interact in real-time.

In the study of electronics and physics, connecting theoretical principles to practical scenarios is key to fostering deeper comprehension and retention of the information.

AC Circuit Analysis

AC circuit analysis involves studying the behavior of alternating current (AC) circuits. It includes examining how components like resistors, inductors, and capacitors interact with each other. Key to this analysis is understanding how these components respond to changes in frequency and voltage.
For an LC circuit, this is crucial when calculating the resonant frequency, as this dictates the point at which the circuit naturally oscillates.

• Calculating impedance, which shows the circuit's opposition to AC flow, varies with frequency.
• Utilizing phasor diagrams helps visualize phase relationships between voltage and current.

Ultimately, grasping AC circuit analysis techniques is vital for electrical engineers and physicists dealing with complex circuit designs and applications.