Sure! Let's first solve the given problem step by step, and then I'll provide two similar problems.

Given Problem

R = 8 ohms, L = 2 mH = 2 × 10^-3 H, frequency (f) = 10,000 Hz

1. Find the capacitance (C) in the circuit:

The damped oscillation frequency $f_d$ in an RLC circuit is given by:

$f_d = \frac{1}{2\pi \sqrt{LC}}$

Rearranging the formula to find $C$:

$C = \frac{1}{(2\pi f_d)^2 L}$

Substituting the known values:

$C = \frac{1}{(2\pi \times 10,000)^2 \times (2 \times 10^{-3})}$

Calculating:

1. $2\pi \times 10,000 \approx 62831.85$
2. $(62831.85)^2 \approx 3.94784 \times 10^9$
3. $C = \frac{1}{3.94784 \times 10^9 \times 2 \times 10^{-3}} \approx \frac{1}{7.89568 \times 10^6} \approx 1.27 \times 10^{-7} \text{ F} = 127 \text{ nF}$

2. What is the critical resistance (R_c) for damped oscillation?

The critical resistance $R_c$ is given by:

$R_c = 2 \sqrt{\frac{L}{C}}$

Substituting the values we calculated:

$R_c = 2 \sqrt{\frac{2 \times 10^{-3}}{1.27 \times 10^{-7}}}$

Calculating:

1. $\frac{2 \times 10^{-3}}{1.27 \times 10^{-7}} \approx 15748.81$
2. $\sqrt{15748.81} \approx 125.5$
3. $R_c = 2 \times 125.5 \approx 251 \text{ ohms}$

Summary of Results

1. Capacitance $C \approx 127 \text{ nF}$
2. Critical Resistance $R_c \approx 251 \text{ ohms}$

Similar Problems

1. Problem 1: In a series RLC circuit, if $R = 5 \text{ ohms}$, $L = 1 \text{ mH}$, and the damped oscillation frequency is $5000 \text{ Hz}$, find the capacitance and the critical resistance.

2. Problem 2: In a parallel RLC circuit, if $R = 10 \text{ ohms}$, $L = 3 \text{ mH}$, and the damped oscillation frequency is $2000 \text{ Hz}$, determine the capacitance and the critical resistance.

Would you like me to solve one of these problems step by step?