To find the required resistance $R$ for the specified decay in an $LC$ circuit, let's analyze the given information.

Problem Details:

• Inductance $L = 217 \, \text{mH} = 0.217 \, \text{H}$
• Capacitance $C = 11.2 \, \mu\text{F} = 11.2 \times 10^{-6} \, \text{F}$
• We want the maximum charge on the capacitor to decay to 99.1% of its initial value over 46.0 cycles.
• The problem assumes that $\omega' \approx \omega$, meaning we can approximate the system’s angular frequency as if it’s an ideal LC circuit.

Step-by-Step Solution

1. Determine the Natural Frequency $\omega$ of the LC Circuit:

   The angular frequency $\omega$ of an LC circuit is given by: $\omega = \frac{1}{\sqrt{LC}}$ Substituting the values for $L$ and $C$: $\omega = \frac{1}{\sqrt{(0.217)(11.2 \times 10^{-6})}}$ Calculate this value to find $\omega$.

2. Determine the Time for 46 Cycles:

   Since $\omega \approx \frac{2 \pi f}{}$ and each cycle corresponds to a period $T = \frac{2\pi}{\omega}$, the time $t$ for 46 cycles is: $t = 46 \cdot T = 46 \cdot \frac{2\pi}{\omega}$

3. Calculate the Decay Requirement Using the Damping Factor:

   For an RLC circuit, the charge $Q$ decays as: $Q(t) = Q_0 e^{-\frac{R}{2L} t}$ Given that the charge decays to 99.1% of its initial value, we set up the equation: $0.991 = e^{-\frac{R}{2L} \cdot t}$ Taking the natural logarithm on both sides: $\ln(0.991) = -\frac{R}{2L} \cdot t$ Solve for $R$: $R = -\frac{2L \cdot \ln(0.991)}{t}$

4. Substitute Known Values and Solve for $R$:

   Substitute the values of $L$ and $t$ (calculated from the 46 cycles) into this equation to find the resistance $R$.

Would you like a detailed calculation for each step, or do you have further questions?

Related Questions

1. How does resistance affect the damping of oscillations in an RLC circuit?
2. Why does the charge decay exponentially in an RLC circuit?
3. What would happen if the resistance $R$ was too high?
4. How can this calculation be used to design circuits with specific damping properties?
5. What is the relationship between frequency and decay in damped oscillatory circuits?

Tip:

Always double-check unit conversions, especially when dealing with millihenries or microfarads, to avoid calculation errors.