# LC circuit with variable capacitor

1. Jul 18, 2012

### pc2-brazil

1. The problem statement, all variables and given/known data
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?

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

3. 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?