Problem 26 - Physics

Question

A series RCL circuit has a resonant frequency of \(690 \mathrm{kHz}\). If the value of the capacitance is \(2.0 \times 10^{-9} \mathrm{~F}\), what is the value of the inductance?

Step-by-Step Solution

Answer

The inductance is approximately \(26.6 \underline{\phantom{xxx}} \mu\text{H}\).

Step 1: Understand the Resonant Frequency Formula

The formula for the resonant frequency \( f \) of a series RCL circuit is given by: \( f = \frac{1}{2\pi \sqrt{LC}} \), where \( L \) is the inductance in Henrys and \( C \) is the capacitance in Farads. Our task is to find \( L \), the inductance.

Step 2: Rearrange the Formula

We need to isolate \( L \) in the resonant frequency formula. Rearranging the equation to solve for \( L \), we get: \( L = \frac{1}{(2\pi f)^2 C} \).

Step 3: Plug in the Given Values

Substitute the given values into the rearranged formula. The given resonant frequency \( f \) is \(690 \text{kHz} = 690 \times 10^3 \text{Hz}\) and the capacitance \( C \) is \(2.0 \times 10^{-9} \text{F}\). The equation becomes: \[ L = \frac{1}{(2\pi \times 690 \times 10^3)^2 \times 2.0 \times 10^{-9}} \].

Step 4: Calculate the Inductance

Perform the calculations: 1. Calculate \( 2\pi \times 690 \times 10^3 \approx 4333796.25 \).2. Square the result: \( (4333796.25)^2 \approx 18781681852828.56 \).3. Multiply by the capacitance: \( 18781681852828.56 \times 2.0 \times 10^{-9} \approx 37563.3637 \).4. Take the reciprocal: \( \frac{1}{37563.3637} \approx 2.66 \times 10^{-5} \text{H} \).

Step 5: Express the Result

The calculated inductance \( L \) is approximately \( 26.6 \underline{\phantom{xxx}} \mu\text{H} \) (microhenries), since \( 1 \underline{\phantom{xxx}} \text{H} = 10^6 \underline{\phantom{xxx}} \mu\text{H} \).

Key Concepts

Resonant Frequency

The concept of resonant frequency is essential in understanding how a series RLC circuit behaves. In these circuits, resonance occurs when the circuit's inductive and capacitive reactances cancel each other out. This happens at a specific frequency called the resonant frequency. At this point, the circuit is purely resistive, and the impedance is at its minimum possible value.

Resonant frequency is determined using the formula: \[ f = \frac{1}{2\pi \sqrt{LC}} \]where:

• \( f \) is the resonant frequency in Hertz (Hz)
• \( L \) is the inductance in Henrys (H)
• \( C \) is the capacitance in Farads (F)

Calculating the resonant frequency helps in designing circuits for applications like radio transmitters and receivers, where tuning to a specific frequency is critical.

Knowing how to manipulate this formula allows engineers to design circuits that perform optimally at desired frequencies.

Inductance Calculation

Calculating the inductance in a resonant circuit is a fundamental task when you know the resonant frequency and the capacitance. By rearranging the resonant frequency formula, you can solve for the inductance \( L \).

The formula becomes:\[ L = \frac{1}{(2\pi f)^2 C}\]This formula shows that the inductance required for a given resonant frequency is inversely proportional to the square of the frequency and directly proportional to the capacitance.

Steps for calculating inductance:

• Convert frequency from kilohertz to hertz if necessary (e.g., \( 690 \text{ kHz} = 690 \times 10^3 \text{ Hz} \)).
• Substitute the given frequency and capacitance values into the formula.
• Calculate \( 2\pi f \) and square it.
• Multiply the result by the capacitance.
• Take the reciprocal to find \( L \).
• Express the result in microhenries if needed, noting that \( 1 \text{ H} = 10^6 \mu\text{H} \).

The process provides the value of inductance in the circuit, crucial for ensuring the system operates efficiently at the specified frequency.

Capacitance in Circuits

Capacitance is a measure of the ability of a system to store an electric charge. In the context of RLC circuits, capacitance plays a crucial role in determining the circuit's resonant frequency. A capacitor in a circuit can affect both the timing and frequency response characteristics.

Capacitors store energy in the electric field between their plates. The impact of capacitance in circuits involves several important aspects:

• **Energy Storage:** Capacitors charge and discharge, allowing circuits to store energy temporarily, which can be essential for energy management solutions.
• **Filtering:** In AC circuits, capacitors block direct current (DC) while allowing alternating current (AC) to pass, making them useful for filtering applications.
• **Frequency Tuning:** Adjusting the capacitance in a circuit can change the resonant frequency, which is valuable in tuning radios or other frequency-dependent devices.

Capacitance, expressed in Farads (F), works hand in hand with inductance to define the characteristics of a resonant circuit. Precision in measuring and using capacitance ensures that circuits fulfill their designed purposes efficiently.