Circuits and resonance

I've done some questions on RLC circuits where I have a second order differential equation of the form: [tex]L\frac{{d^2 q}}{{dt^2 }} + R\frac{{dq}}{{dt}} + \frac{q}{C} = E\left( t \right)[/tex]

The solution of this equation gives an expression for the charge as a function of time, q(t). Just off the top of my head I think that the numbers(for the inductance, resistance etc) can be fudged so that the particular solution is something of the form [tex]q\left( t \right) = Rt\cos \left( {\omega t + \varphi } \right)[/tex]. In other words, the expression for long term behaviour of q(t) is a series of unbounded oscillations.

I'm not sure if this sort of thing actually happens in real life, but the math suggests that 'resonance' occurs. And I'm thinking that at least some of the 'applications' questions I've been doing have some relation to real life. I'd basically like to know if there is any physical interpretation of a q(t) with unbounded oscillations? Charge appearing from out of nowhere? I'm pretty clueless when it comes to interpreting the math. So can someone shed some light on this problem for me?


Staff Emeritus
Science Advisor
Take a look at this, which has the equation in terms of current rather than charge -

E(t) would be a forcing function, which induces a 'forced' oscillation. The left hand side may produce a damped, overdamped or underdamped condition regarding oscillation, depending on R, L and C.

Charges do not appear out of nowhere, unless there is some particle interaction like pair production, and in that cases, charge neutrality is still maintained.

In a wire bearing AC current, the charges (electrons) are simply moving back and forth under the influence of an applied emf.

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