What resistance R should be connected in series with an...

What resistance $R$ should be connected in series with an inductance $L=220 \mathrm{mH}$ and capacitance $C=12.0 \mu \mathrm{F}$ for the maximum charge on the capacitor to decay to 99.0$\%$ of its initial value in 50.0 cycles? (Assume $\omega^{\prime} \approx \omega . )$

Key Concepts

Damped Harmonic Oscillator

This concept refers to any oscillatory system in which energy is gradually lost over time due to a damping force. In the context of an RLC (resistor-inductor-capacitor) circuit, resistance causes the amplitude of the oscillations (such as the charge on a capacitor) to decay exponentially. Understanding damped oscillators is crucial for analyzing how quickly the system’s energy decays and for determining the parameters that control this decay.

Exponential Decay in Oscillatory Systems

Exponential decay describes the process where the amplitude of the oscillations decreases by a constant fraction in each time interval. For lightly damped oscillators, the envelope of the oscillation decays as an exponential function of time. This concept is used to relate the decay rate to the circuit parameters (like resistance and inductance) and is fundamental for calculations involving the duration or number of cycles needed for the amplitude to drop to a specific level.

Natural Frequency of LC Circuits

The natural frequency is the rate at which an LC circuit would oscillate if there were no resistance (and hence, no damping). It is given by ? = 1/?(LC) and determines the period of the oscillations. In a damped system where the damping is light, this frequency remains approximately the same for several cycles. Knowing the natural frequency is essential for linking time intervals (such as cycles) to the overall decay process.

Logarithmic Decrement

Logarithmic decrement is a measure of the rate of decay of oscillations over a given number of cycles. It quantifies the natural logarithm of the ratio of successive amplitudes and is commonly used to connect the change in amplitude to the damping coefficient. This concept is key when determining how parameters like resistance affect the exponential decay of amplitude over multiple cycles in an RLC circuit.

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