LC Circuit
- Claimed by Rishab Chawla 11/19/15***
Main Idea
Consider an electrical circuit consisting of an inductor, of inductance L, connected in series with a capacitor, of capacitance C. Such a circuit is known as an LC circuit, for obvious reasons.
A Mathematical Model
Starting with Kirchoff's Loop Rule, we have V = L * dI/dt + q/C.
Taking the derivative of each term, dV/dt = L(d^2*I/dt^2) + 1/C * dq/dt.
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(d^2*I/dt^2) + 1/C * I
Rearranging, (d^2/dt^2)I = - I/(LC).
This can be solved by guessing and checking with a generic sine function: I = I0sin(ωt + φ)
(d^2/dt^2)I0sin(ωt + φ) = -I0/(LC) * sin(ωt + φ) − ω^2*I0sin(ωt + φ) = -I0/(LC) * sin(ωt + φ)
The angular frequency is now given by ω = 1/sqrt(LC). A LC circuit is then an oscillating circuit with frequency ω/(2π) = 1/(2π*sqrt(LC)).
A Computational Model
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