The frequency of free electromagnetic oscillations in the oscillatory circuit is 20 MHz.

The frequency of free electromagnetic oscillations in the oscillatory circuit is 20 MHz. Determine the frequency of oscillations in the circuit if you replace a capacitor with a capacity of 1 μF with a capacitor with a capacitance of 4 μF.

To determine the value of the oscillation frequency after increasing the capacitance of the capacitor, we apply the formula: ν = 1 / (2 * Π * √ (L * C)), whence L * C = 1 / (ν ^ 2 * 2 * Π); C * ν ^ 2 = 1 / (L * 4 * Π ^ 2). It follows that one can use the equality: C1 * ν1 ^ 2 = 1 / (L * 4 * Π ^ 2) = C2 * ν2 ^ 2, whence ν2 = √ (C1 * ν1 ^ 2 / C2).

Values ​​of variables: C1 – capacity of the initial capacitor (C1 = 1 μF = 1 * 10 ^ -6 F); ν1 – initial vibration frequency (ν1 = 20 MHz = 20 * 10 ^ 6 Hz); C2 is the capacity of the final capacitor (C2 = 4 μF = 4 * 10 ^ -6 F).

Calculation: ν2 = √ (C1 * ν1 ^ 2 / C ^ 2) = √ (1 * 10 ^ -6 * (20 * 10 ^ 6) ^ 2 / (4 * 10 ^ -6)) = 10 * 10 ^ 6 Hz = 10 MHz.

Answer: The oscillation frequency has decreased to 10 MHz.