Begin with Kirchhoff's circuit rule.
|V = L||dI||+||q|
Take the derivative of each term.
The voltage of the battery is constant, so that derivative vanishes. The derivative of charge is current, so that gives us a second order differential equation.
|0 = L||d2I||+||1||I|
Rearrange it a bit…
|d2||I = −||1||I|
and then pause to consider a solution.
We need a function whose second derivative is itself with a minus sign. We have two options: sine and cosine. Either one is fine since they're basically identical functions with a 90° phase shift between them. Without loss of generality, I'll choose sine with an arbitrary phase angle (φ) that could equal 90° if we let it. Or it could be equal to some other angle. The other parameters in a generic sine function are amplitude (I0) and angular frequency (ω).
The basic method I've started is called "guess and check". My guess is that the function looks like a generic sine function…
I = I0 sin(ωt + φ)
and the check is to pop it back into the differential equation and see what happens.
|d2||I0 sin(ωt + φ)||= −||1||I0 sin(ωt + φ)|
|− ω2I0 sin(ωt + φ)||= −||1||I0 sin(ωt + φ)|
Basically everything cancels but one parameter — angular frequency.
An LC circuit is therefore an oscillating circuit. The frequency of such a circuit (as opposed to its angular frequency) is given by…
So what? How is this useful?
An audio crossover circuit consisting of three LC circuits, each tuned to a different natural frequency is shown to the right. The inductors (L) are on the top of the circuit and the capacitors (C) are on the bottom. On the left a "woofer" circuit tuned to a low audio frequency, on the right a "tweeter" circuit tuned to a high audio frequency, and in between a "midrange" circuit tuned to a frequency in the middle of the audio spectrum.
RC circuits are basically filters.
I need to write this part.