# LC circuit with variable capacitor

• pc2-brazil
In summary, an oscillating LC circuit with a maximum current of 31 mA and fixed inductance of 42 mH is designed to operate with a variable capacitor C. The circuit must be operated safely at a frequency of 1.0 MHz and the highest frequency that won't damage the capacitor must be determined. The minimum capacitance is also asked for. Using the equations for oscillatory frequency and maximum magnetic energy stored in the inductor, the solution is found to be 16 nF for minimum capacitance and 6.1 kHz for maximum frequency. However, there is a discrepancy with the answer in the book, which is 5717.2 Hz. Further checking of the calculations is recommended.
pc2-brazil

## Homework Statement

An oscillating LC circuit is designed to operate with a maximum current of 31 mA. The inductance is fixed at 42 mH, and the frequency is changed by means of a variable capacitor C. (a) If the highest voltage that the capacitor can handle is 50 V, can the circuit be operated safely at the frequency of 1.0 MHz? (b) What is the highest frequency of operation that doesn't damage the capacitor? (c) What is the minimum capacitance?

## Homework Equations

Oscillatory frequency of an LC circuit:
$$\nu = \frac{1}{2\pi} \sqrt{\frac{1}{LC}}$$

## The Attempt at a Solution

The maximum magnetic energy stored in the inductor is Li²/2, where i is the maximum current. This is equal in value to the maximum electric energy that will be stored in the capacitor, therefore Li²/2 = CV²/2.
Since the maximum current is 0.031 A, the maximum magnetic energy stored in the inductor is Li²/2 = (0.042)(0.031)²/2 = 0.000020181 = 20.181 μJ. Since the maximum voltage is 50 V, the minimum capacitance is given by CV²/2 = Li²/2, so C(50)²/2 = 0.000020181. This gives 16 nF, which is correct, according to the book.
My problem here is to find the maximum frequency. Frequency is given by:
$$\nu = \frac{1}{2\pi} \sqrt{\frac{1}{LC}}$$
The maximum frequency should be the value above when C is the value of the minimum capacitance; this gives 5717.2 Hz. But the answer in the back of the book is 6.1 kHz.

What am I doing wrong?

Thank you in advance.

Hello. Recheck you calculation of the frequency. If I use 16 nF and .042 H, I get 6.1 kHz.

## 1. What is an LC circuit with a variable capacitor?

An LC circuit with a variable capacitor is a type of electrical circuit that consists of an inductor (L) and a capacitor (C) connected in parallel. The capacitor in this circuit is a variable capacitor, meaning its capacitance can be changed. This allows for the resonant frequency of the circuit to be adjusted and tuned to a specific desired frequency.

## 2. How does an LC circuit with a variable capacitor work?

An LC circuit with a variable capacitor works by storing electrical energy in the capacitor and magnetic energy in the inductor. As the capacitor's capacitance is changed, the resonant frequency of the circuit also changes. This causes the circuit to oscillate at the resonant frequency, which can be used for various applications such as signal filtering and tuning in radio communication.

## 3. What is the resonant frequency of an LC circuit with a variable capacitor?

The resonant frequency of an LC circuit with a variable capacitor is determined by the formula: f = 1/(2π√(LC)). This frequency can be adjusted by changing the capacitance of the variable capacitor in the circuit.

## 4. What are some real-world applications of an LC circuit with a variable capacitor?

LC circuits with variable capacitors are commonly used in radio communication to tune and filter signals. They are also used in electronic circuits for frequency-selective filters, oscillators, and electronic tuners. In addition, they can be found in electronic devices like TVs and radios for channel selection and tuning.

## 5. What are the advantages of using an LC circuit with a variable capacitor?

The main advantage of using an LC circuit with a variable capacitor is its ability to tune to a specific desired frequency. This makes it useful in many different applications, such as signal filtering and tuning. Additionally, these circuits are relatively simple and inexpensive to construct, making them a popular choice in electronic circuits.

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