Ch. 30 - Inductance, Electromagnetic Oscillations, and AC Circuits

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 30 - Inductance, Electromagnetic Oscillations, and AC Circuits

Problem 72a

Chapter 29, Problem 72a

For an underdamped LRC circuit, determine a formula for the energy U = UE + UB stored in the electric and magnetic fields as a function of time. Give answer in terms of the initial charge Q_o on the capacitor.

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Start by recalling the total energy stored in an LRC circuit, which is the sum of the energy stored in the electric field of the capacitor $ U_E $ and the energy stored in the magnetic field of the inductor $ U_B $. The total energy is given by $ U = U_E + U_B $.

The energy stored in the capacitor is $ U_E = \frac{1}{2} C V^2 $, where $ C $ is the capacitance and $ V $ is the voltage across the capacitor. Since $ V = \frac{Q}{C} $, this can be rewritten as $ U_E = \frac{1}{2} \frac{Q^2}{C} $, where $ Q $ is the charge on the capacitor.

The energy stored in the inductor is $ U_B = \frac{1}{2} L I^2 $, where $ L $ is the inductance and $ I $ is the current through the inductor. Using the relationship $ I = -\frac{dQ}{dt} $, this becomes $ U_B = \frac{1}{2} L \left( \frac{dQ}{dt} \right)^2 $.

For an underdamped LRC circuit, the charge on the capacitor as a function of time is given by $ Q(t) = Q_0 e^{-\gamma t} \cos(\omega_d t) $, where $ Q_0 $ is the initial charge, $ \gamma = \frac{R}{2L} $ is the damping coefficient, and $ \omega_d = \sqrt{\frac{1}{LC} - \gamma^2} $ is the damped angular frequency. Substitute this expression for $ Q(t) $ into the formulas for $ U_E $ and $ U_B $.

Combine $ U_E $ and $ U_B $ to express the total energy $ U(t) $ as a function of time. Simplify the expression to show how the energy decays over time due to the damping factor $ e^{-2\gamma t} $. The final formula will involve $ Q_0 $, $ C $, $ L $, $ \gamma $, and $ \omega_d $.

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Key Concepts

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

Underdamped LRC Circuit

An underdamped LRC circuit is a type of electrical circuit that consists of an inductor (L), a resistor (R), and a capacitor (C) where the resistance is low enough that the system oscillates rather than quickly settling to equilibrium. In this scenario, the circuit exhibits oscillatory behavior with a gradually decreasing amplitude, allowing for the analysis of energy transfer between the electric and magnetic fields over time.

Energy in Electric and Magnetic Fields

In an LRC circuit, energy is stored in two forms: electric energy in the capacitor (UE) and magnetic energy in the inductor (UB). The electric energy is given by UE = 1/2 * C * V^2, where V is the voltage across the capacitor, while the magnetic energy is given by UB = 1/2 * L * I^2, where I is the current through the inductor. The total energy U = UE + UB varies over time as energy oscillates between these two forms.

Initial Charge (Qo) on the Capacitor

The initial charge Qo on the capacitor is a critical parameter that influences the behavior of the LRC circuit. It determines the initial voltage across the capacitor and, consequently, the initial energy stored in the electric field. As the circuit oscillates, this initial charge plays a significant role in calculating the time-dependent energy stored in both the electric and magnetic fields, allowing for the derivation of a formula for U as a function of time.

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Textbook Question

At t = 0, the current through a 60.0-mH inductor is 50.0 mA and is increasing at the rate of 78.0 mA/s. What is the initial energy stored in the inductor, and how long does it take for the energy to increase by a factor of 8.0 from the initial value?

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Textbook Question

An ac voltage source V = Vo sin (ωt + 90°) is connected across an inductor L and current I = Io sin (ωt) flows in this circuit. Note that the current and source voltage are 90° out of phase.

(a) Directly calculate the average power delivered by the source over one period T of its sinusoidal cycle via the integral P = ∫₀ᵀ VI dt/T.

(b) Apply the relation P = Iᵣₘₛ Vᵣₘₛ cos Φ to this circuit and show that the answer you obtain is consistent with that found in part (a). Comment on your results.

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Textbook Question

An inductance coil draws 2.2 A dc when connected to a 45-V battery. When connected to a 60.0-Hz 120-V (rms) source, the current drawn is 3.8 A (rms). Determine the inductance and resistance of the coil.

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Textbook Question

At time t = 0, the switch in the circuit shown in Fig. 30–30 is closed. After a sufficiently long time, steady currents I₁, I₂, and I₃ flow through resistors R₁, R₂, and R₃, respectively. Determine these three currents.

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Textbook Question

A pair of straight parallel thin wires, such as a lamp cord, each of radius r, are a distance 𝓁 apart and carry current to a circuit some distance away. Ignoring the field within each wire, show that the inductance per unit length is (μ₀/π) ln[(𝓁 - r) /r].

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Textbook Question

Show that the fraction of electromagnetic energy lost (to thermal energy) per cycle in a lightly damped (R² ≪ 4L/C) LRC circuit is approximately $\frac{Δ U}{U} = \frac{2 π R}{L ω} = \frac{2 π}{Q}$ . The quantity Q can be defined as Q = Lω/R, and is called the Q-value, or quality factor, of the circuit and is a measure of the damping present. A high Q-value means smaller damping and less energy input required to maintain oscillations.

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For an underdamped LRC circuit, determine a formula for the energy U = UE + UB stored in the electric and magnetic fields as a function of time. Give answer in terms of the initial charge Qo on the capacitor.

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Key Concepts

Underdamped LRC Circuit

Energy in Electric and Magnetic Fields

Initial Charge (Qo) on the Capacitor

For an underdamped LRC circuit, determine a formula for the energy U = UE + UB stored in the electric and magnetic fields as a function of time. Give answer in terms of the initial charge Q_o on the capacitor.