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 84a

Chapter 29, Problem 84a

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.

Verified step by step guidance

Start by recalling the formula for the angular frequency of an LC circuit: ω = 1 / √(LC), where L is the inductance and C is the capacitance of the circuit.

The capacitance of a parallel-plate capacitor is given by the formula C = (ε₀A) / x, where ε₀ is the permittivity of free space, A is the area of the plates, and x is the separation between the plates.

Substitute the expression for C into the angular frequency formula: ω = 1 / √(L * (ε₀A / x)).

Rearrange the equation to isolate x. Start by squaring both sides to eliminate the square root: ω² = 1 / (L * (ε₀A / x)). Then multiply through by (L * (ε₀A / x)) and solve for x: x = 1 / (ω²Lε₀A).

Finally, express ω in terms of the given frequency f using the relationship ω = 2πf. Substitute this into the equation for x: x = 1 / ((2πf)²Lε₀A). Simplify to get the final expression: x = 4π²Aε₀f²L.

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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 electric charge per unit voltage. In a parallel-plate capacitor, it is determined by the area of the plates (A), the distance between them (x), and the permittivity of the material between the plates (ε₀). The formula for capacitance is C = ε₀A/x, which is crucial for understanding how changes in plate separation affect the circuit's behavior.

Inductance

Inductance is a property of an electrical circuit that quantifies the ability to store energy in a magnetic field when current flows through it. In an LC circuit, inductance (L) works in conjunction with capacitance (C) to determine the oscillation frequency (f) of the circuit. The relationship between inductance and frequency is essential for analyzing the dynamics of the circuit.

Oscillation Frequency

The oscillation frequency (f) in an LC circuit is the rate at which the charge on the capacitor oscillates, and it is given by the formula f = 1/(2π√(LC)). This frequency is influenced by both the inductance and capacitance of the circuit. Understanding how frequency relates to the physical parameters of the circuit is key to deriving the relationship between distance (x) and other variables in the problem.

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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 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.

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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

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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