What is Natural period and frequency? — Physics

Definition 31.3 High School Physics · Chapter 31 — Free Electrical Oscillations: RLC

The natural period of an LC circuit is $T_0 = 2\pi\sqrt{LC}$ ; its natural frequency is

f_0 = \frac{1}{T_0} = \frac{1}{2\pi\sqrt{LC}} .

Dimensional check: $1\,\mathrm{H} = 1\,\mathrm{V}\,\mathrm{s}/\mathrm{A}$ and $1\,\mathrm{F} = 1\,\mathrm{C}/\mathrm{V}$ , so $LC$ carries $\mathrm{s} \times \mathrm{C}/\mathrm{A} = \mathrm{s}^{2}$ — $\sqrt{LC}$ is a time, as it must be.

Charge and current of the free LC circuit: the current peaks each time the charge crosses zero, a quarter period out of step.

Examples

Example 31.4 (First numbers)

$L = 10\,\mathrm{mH}$ with $C = 1.0\,\text{µ}\mathrm{F}$ : $\sqrt{LC} = 1.0 \times 10^{-4}\,\mathrm{s}$ , so $T_0 = 0.63\,\mathrm{ms}$ and $f_0 \approx 1.6\,\mathrm{kHz}$ — an audible tone. $L = 1.0\,\mathrm{mH}$ with $C = 100\,\mathrm{pF}$ : $T_0 = 2.0\,\text{µ}\mathrm{s}$ , $f_0 \approx 0.50\,\mathrm{MHz}$ — a radio frequency. Only the product $LC$ sets the tempo.

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