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 84c

Chapter 29, Problem 84c

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 f is on the order of 1 MHz and can be measured to a precision of ∆f = 1 Hz, with what percent accuracy can x be determined? Assume fringing effects at the capacitor’s edges can be neglected.

Verified step by step guidance

Start by understanding the relationship between the resonant frequency $ f $ of an LC circuit and the capacitance $ C $. The resonant frequency is given by the formula: $ f = \frac{1}{2 \pi \sqrt{L C}} $, where $ L $ is the inductance.

Rearrange the formula to solve for the capacitance $ C $: $ C = \frac{1}{(2 \pi f)^2 L} $. This shows how the capacitance depends on the resonant frequency $ f $ and the inductance $ L $.

The capacitance $ C $ of a parallel-plate capacitor is related to the plate area $ A $, the separation distance $ x $, and the permittivity of free space $ \varepsilon_0 $ by the formula: $ C = \frac{\varepsilon_0 A}{x} $. Rearrange this to solve for $ x $: $ x = \frac{\varepsilon_0 A}{C} $.

Substitute the expression for $ C $ from the resonant frequency formula into the equation for $ x $: $ x = \varepsilon_0 A (2 \pi f)^2 L $. This relates the separation distance $ x $ to the measurable frequency $ f $, the inductance $ L $, and the plate area $ A $.

To determine the percent accuracy of $ x $, use the relationship between the uncertainty in $ f $ ($ \Delta f $) and the uncertainty in $ x $. Since $ x $ depends on $ f^2 $, the relative uncertainty in $ x $ is approximately twice the relative uncertainty in $ f $: $ \frac{\Delta x}{x} \approx 2 \frac{\Delta f}{f} $. Multiply this by 100 to express the accuracy as a percentage.

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Key Concepts

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

Capacitance

Capacitance is the ability of a system to store an electric charge, defined as the ratio of the electric charge on each conductor to the potential difference between them. In a parallel-plate capacitor, capacitance (C) is given by the formula C = ε₀(A/d), where ε₀ is the permittivity of free space, A is the area of the plates, and d is the separation between them. Understanding capacitance is crucial for analyzing how changes in distance between the plates affect the circuit's behavior.

LC Circuit

An LC circuit is an electrical circuit consisting of an inductor (L) and a capacitor (C) connected together. It can oscillate at a natural resonant frequency, which is determined by the values of L and C. The frequency of oscillation (f) is given by the formula f = 1/(2π√(LC)). This concept is essential for understanding how the frequency of the circuit relates to the physical parameters of the capacitor and inductor, particularly in the context of measuring tiny distances.

Frequency Measurement Precision

Frequency measurement precision refers to the smallest change in frequency that can be reliably detected, denoted as ∆f. In this context, if the frequency f is measured with a precision of ∆f = 1 Hz, it implies that any variations in the circuit's parameters, such as the distance between capacitor plates, can be inferred from changes in frequency. The percent accuracy in determining a physical quantity, like distance, can be derived from the relationship between frequency and the parameters of the LC circuit.

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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

Show that if the inductor L in the filter circuit of Fig. 30–33 (Problem 87) is replaced by a large resistor R, there will still be significant attenuation of the ac voltage and little attenuation of the dc voltage if the input dc voltage is high and the current (and power) are low.

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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

Filter circuit. Figure 30–33 shows a simple filter circuit designed to pass dc voltages with minimal attenuation and to remove, as much as possible, any ac components (such as 60-Hz line voltage that could cause hum in an audio system, for example). Assume V_in = V₁ + V₂ where V₁ is dc and V₂ = V₂₀ sin ωt, and that any resistance is very small. (a) Determine the current through the capacitor: give amplitude and phase (assume R = 0 and X_L > X_C). (b) Show that the ac component of the output voltage, V_2out, equals (Q/C) - V₁ where Q is the charge on the capacitor at any instant, and determine the amplitude and phase of V_2out (c) Show that the attenuation of the ac voltage is greatest when X_C << X_L, and calculate the ratio of the output to input ac voltage in this case. (d) Compare the dc output voltage to input voltage.

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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

Suppose a series LRC circuit has two resistors, R₁ and R₂, two capacitors, C₁ and C₂, and two inductors, L₁ and L₂ all in series. Calculate the total impedance of the circuit.

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