(II) (a) Show that oscillation of charge Q on the capacitor...

(II) (a) Show that oscillation of charge $Q$ on the capacitor of an $L R C$ circuit has amplitude $Q _ { 0 } = \frac { V _ { 0 } } { \sqrt { ( \omega R ) ^ { 2 } + \left( \omega ^ { 2 } L - \frac { 1 } { C } \right) ^ { 2 } } }$ (b) At what angular frequency, $\omega ^ { \prime } ,$ will $Q _ { 0 }$ be a maximum? $( c )$ Compare to a forced damped harmonic oscillator, and discuss. (See Question 20 in this Chapter.)

Key Concepts

Forced Oscillations in LRC Circuits

This concept involves analyzing the behavior of an LRC circuit when it is driven by an external alternating voltage source. It requires setting up the differential equation that governs the charge or current, which includes contributions from the inductor, resistor, and capacitor. The steady state (or forced response) of the circuit is of particular interest because it represents how the system responds at a particular driving frequency, leading to expressions that describe amplitude and phase of the oscillations.

Complex Impedance and Phasor Representation

In AC circuit analysis, components such as resistors, inductors, and capacitors are represented using complex impedance, which simplifies the handling of the phase relationships between voltage and current. Phasor representation allows these sinusoidally varying quantities to be treated as complex numbers, making it easier to derive the amplitude of response by handling the reactive (imaginary) and resistive (real) parts simultaneously. This approach yields formulas like the one for the oscillation amplitude in the forced LRC circuit.

Resonance in RLC Circuits

Resonance is the phenomenon where the system oscillates with maximum amplitude at a particular driving frequency, typically when the reactive effects of the inductor and capacitor cancel each other out. In an LRC circuit, resonance occurs when the inductive reactance equals the capacitive reactance, leading to a simplified impedance that is only real (resistive). Identifying the frequency at which this cancellation occurs is crucial for maximizing the amplitude of oscillations, and is directly related to the expression showing a maximum amplitude.

Forced Damped Harmonic Oscillator Analogy

The forced damped harmonic oscillator is a classical mechanical system subject to a driving force and damping, and it shares a mathematical form with the LRC circuit equation. This analogy helps in understanding the behavior of the RLC circuit, particularly in terms of resonance, phase shift, and the dependence of amplitude on the driving frequency. Through this comparison, one can draw parallels between the damping effects in a mechanical oscillator due to friction and the resistive losses in an electrical circuit.

Transcript

00:01 So for the rlc circuit in oscillation, the phaser for q lags behind the phaser for i by 90 degrees, and the amplitude is related to the amplitude of i by q0 equals i0 divided by omega.

00:30 This is simply because i equals dqdt.

00:39 So if q is some trigonometry function, then it will bring down effect of omega for i.

00:55 So therefore, q0 equals i0 over omega, and i0 equals v0 divided by the impotence.

01:14 And for the r ilc circuit, the impedance is given by the square root of r squared plus, omega l minus 1 over omega c squared.

01:35 And by sending this omega into the square root, we get e not over square root omega squared r squared plus omega square l minus 1 over c squared.

01:56 So this is indeed the form given by the problem.

01:59 And next, in order to find the maximum of q0, let's expand the polynomial inside the square root.

02:17 So this will give us v0 over l square omega to the fourth plus.

02:43 And we can view this polynomial as a quadratic polynomial of the variable omega squared.

02:53 And by completing the square, we can rewrite.

02:59 This polynomial as square root of minus so if 1 over lc is bigger than r square over 2 l squared q not has maximum at omega squared equals 1 over lc minus r squared minus over 2 l squared so when omega prime equals square root of 1 over lc minus r squared over 2 l squared q not reaches its maximum at e knot divided by r times squared root of 1 over lc minus r squared over 4 l squared and if 1 over lc is smaller than r squared over 2 l squared then at this time, q0 reaches maximum at omega prime equals 0...

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