An LCR circuit is equivalent to a damped pendulum. In an LCR...

An LCR circuit is equivalent to a damped pendulum. In an LCR circuit, the capacitor is charged to $Q_{0}$ and then connected to the $L$ and $R$ as shown below If a student plots graphs of the square of maximum charge $\left(Q_{\mathrm{Max}}^{2}\right)$ on the capacitor with time $(t)$ for two different values $L_{1}$ and $L_{2}\left(L_{1}>L_{2}\right)$ of $L$ then which of the following represents this graph correctly? (Plots as schematic and not drawn to scale) (A) (B) (C) (D)

Key Concepts

Effect of Inductance on Damping

Within an LCR circuit, the value of the inductance affects the rate of amplitude decay. A higher inductance results in a slower decay (less damping) because the damping factor is inversely proportional to L. Consequently, when comparing circuits with different inductance values, the circuit with the larger L exhibits a more gradual decline in the squared amplitude as a function of time.

Exponential Decay Envelope

The maximum amplitude in a damped oscillation decays exponentially with time. In an LCR circuit, if the capacitor initially holds a charge Q0, the amplitude of the charge oscillations decreases roughly as Q0 exp(-Rt/(2L)) for an underdamped system. Squaring the amplitude results in an exponential decay profile whose time constant depends on the system’s parameters.

LCR Circuit Oscillations

An LCR circuit (comprising an inductor, capacitor, and resistor) exhibits oscillatory behavior analogous to mechanical oscillators such as a pendulum or mass?spring system. Energy alternates between the inductor (magnetic field) and capacitor (electric field), while the resistor causes energy dissipation, leading to damping of the oscillations.

Damped Oscillations

In any damped system, including electrical and mechanical oscillators, the amplitude of oscillations decreases over time. The presence of a resistive element in an LCR circuit, or friction in a mechanical system, gives rise to energy loss that causes the oscillation envelope to decay exponentially.

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