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 81

Chapter 29, Problem 81

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.

Verified step by step guidance

Step 1: Understand the problem. The coil has both resistance (R) and inductance (L). When connected to a DC source, only the resistance affects the current. When connected to an AC source, both resistance and inductive reactance (X_L) affect the current. Use this information to set up equations for R and L.

Step 2: Use Ohm's Law for the DC circuit. The current in the DC circuit is given by I = V / R. Rearrange this to find the resistance: R = V / I, where V = 45 V and I = 2.2 A.

Step 3: For the AC circuit, calculate the total impedance (Z). The relationship between voltage, current, and impedance in an AC circuit is given by V_rms = I_rms * Z. Rearrange to find Z: Z = V_rms / I_rms, where V_rms = 120 V and I_rms = 3.8 A.

Step 4: Relate impedance (Z), resistance (R), and inductive reactance (X_L). In an AC circuit, the impedance is given by Z = √(R² + X_L²). Use the value of R from Step 2 and the value of Z from Step 3 to solve for X_L: X_L = √(Z² - R²).

Step 5: Calculate the inductance (L) using the formula for inductive reactance: X_L = 2πfL, where f = 60.0 Hz. Rearrange to find L: L = X_L / (2πf). Use the value of X_L from Step 4 to determine L.

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Key Concepts

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

Inductance

Inductance is a property of an electrical component, typically a coil, that quantifies its ability to store energy in a magnetic field when an electric current flows through it. It is measured in henries (H) and is crucial in AC circuits, as it affects how the current and voltage relate to each other over time, particularly in response to changes in current.

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. This fundamental relationship is expressed as V = IR, and it is essential for analyzing both DC and AC circuits, allowing for the calculation of current, voltage, or resistance when the other two quantities are known.

RMS Current and Voltage

RMS (Root Mean Square) values are used to express the effective value of alternating current (AC) and voltage. The RMS value of an AC signal is equivalent to a DC value that would deliver the same power to a load. Understanding RMS is vital for analyzing AC circuits, as it allows for the comparison of AC and DC quantities and is used in calculations involving power and impedance.

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RMS Current and Voltage

Related Practice

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

In some experiments, very tiny distances or spaces ( ≈ nm ) can be measured by using capacitance. Consider forming an LC circuit using a parallel-plate capacitor with plate area A, and a known inductance L. If charge is found to oscillate in this circuit at frequency f = ω/2π when the capacitor plates are separated by distance x, show that x = 4π² Aε₀f²L.

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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

In some experiments, very tiny distances or spaces ( ≈ nm ) can be measured by using capacitance. Consider forming an LC circuit using a parallel-plate capacitor with plate area A, and a known inductance L. When the plate separation is changed by ∆x, the circuit’s oscillation frequency will change by ∆f. Show that ∆x/x ≈ 2(∆f/f).

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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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Textbook Question

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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