An L-R-C series circuit has L = 0.600 H and C = 3.00 mF. (b) What value of R gives critical damping?

Verified step by step guidance

1. Identify the formula for critical damping in an L-R-C series circuit. The critical damping resistance (R) can be calculated using the formula: R = 2 \sqrt{\frac{L}{C}}.
2. Substitute the given values of L (inductance) and C (capacitance) into the formula. Here, L = 0.600 H and C = 3.00 mF. Note: Convert the capacitance from millifarads to farads by using the conversion 1 mF = 1 \times 10^{-3} F.
3. Perform the calculation under the square root first, which involves dividing the inductance by the capacitance.
4. Take the square root of the result obtained in the previous step.
5. Multiply the square root value by 2 to find the value of R that results in critical damping.

Key Concepts

Critical Damping

Critical damping occurs in a system when the damping force is just enough to prevent oscillation, allowing the system to return to equilibrium in the shortest time without overshooting. In an L-R-C circuit, critical damping is achieved when the resistance (R) is set to a specific value that balances the inductance (L) and capacitance (C) of the circuit, preventing oscillatory behavior.

Damping Ratio

The damping ratio is a dimensionless measure that describes how oscillations in a system decay after a disturbance. It is defined as the ratio of the actual damping to the critical damping. For an L-R-C circuit, the damping ratio helps determine whether the circuit will be underdamped, overdamped, or critically damped based on the values of R, L, and C.

Resonant Frequency

Resonant frequency is the frequency at which a system naturally oscillates when not subjected to a damping force. In an L-R-C circuit, the resonant frequency is determined by the inductance (L) and capacitance (C) and is given by the formula ω₀ = 1/√(LC). Understanding resonant frequency is essential for analyzing the behavior of the circuit, especially in relation to damping and oscillations.