Ch 30: Inductance

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 30: Inductance

Problem 38b

Chapter 30, Problem 38b

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

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Understand that critical damping in an L-R-C circuit occurs when the damping is just enough to prevent oscillations. This happens when the damping coefficient is equal to the natural frequency of the system.

Recall the formula for the damping coefficient \( \gamma \) in an L-R-C circuit, which is given by \( \gamma = \frac{R}{2L} \).

The natural frequency \( \omega_0 \) of the circuit is given by \( \omega_0 = \frac{1}{\sqrt{LC}} \).

For critical damping, set the damping coefficient equal to the natural frequency: \( \frac{R}{2L} = \frac{1}{\sqrt{LC}} \).

Solve for \( R \) by rearranging the equation: \( R = 2L \times \frac{1}{\sqrt{LC}} \). Substitute the given values of \( L = 0.600 \) H and \( C = 3.00 \) mF to find the value of \( R \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Critical Damping

Critical damping occurs in a system when the damping is just enough to prevent oscillations. In an L-R-C circuit, it ensures the system returns to equilibrium as quickly as possible without oscillating. It is achieved when the damping coefficient equals the natural frequency of the system, leading to a smooth and rapid return to equilibrium.

Damping in L-R-C Circuits

Damping in L-R-C circuits refers to the resistance that reduces the amplitude of oscillations over time. It is influenced by the resistor (R) in the circuit. The damping effect can be underdamped, critically damped, or overdamped, depending on the value of R relative to the inductance (L) and capacitance (C). Critical damping is the optimal condition for rapid stabilization.

Resonance Frequency

The resonance frequency in an L-R-C circuit is the frequency at which the circuit naturally oscillates when not damped. It is determined by the inductance (L) and capacitance (C) and is given by the formula ω₀ = 1/√(LC). This frequency is crucial for calculating the damping conditions, including critical damping, as it relates to the system's natural response.

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Related Practice

Textbook Question

An L-R-C series circuit has L = 0.600 H and C = 3.00 mF. Calculate the angular frequency of oscillation for the circuit when R = 0.

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Textbook Question

An L-R-C series circuit has L = 0.450 H, C = 2.50 × 10^-5 F, and resistance R. What is the angular frequency of the circuit when R = 0?

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Textbook Question

An L-R-C series circuit has L = 0.450 H, C = 2.50 × 10^-5 F, and resistance R. What value must R have to give a 5.0% decrease in angular frequency compared to the value calculated in part (a)?

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Textbook Question

It has been proposed to use large inductors as energy storage devices. How much electrical energy is converted to light and thermal energy by a 150 W light bulb in one day?

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Textbook Question

It has been proposed to use large inductors as energy storage devices. If the amount of energy calculated in part is stored in an inductor in which the current is 80.0 A, what is the inductance?

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