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Ch. 31 - Maxwell's Equations and Electromagnetic Waves
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 31 - Maxwell's Equations and Electromagnetic WavesProblem 6c
Chapter 30, Problem 6c

Suppose an air-gap capacitor has circular plates of radius r = 2.5 cm and separation d = 1.6 mm. A 68.0-Hz emf, ε = ε₀ cos ωt, is applied to the capacitor. The maximum displacement current is 35 μA. Determine the maximum value of dΦE/dt between the plates. Neglect fringing.

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The displacement current (I_d) is related to the rate of change of the electric flux (dΦ_E/dt) by the equation: Id=ε0dΦEdt, where ε₀ is the permittivity of free space (8.85 × 10⁻¹² F/m). Rearrange this equation to solve for dΦ_E/dt: dΦEdt=Idε0.
Substitute the given values into the equation. The displacement current is I_d = 35 μA = 35 × 10⁻⁶ A, and ε₀ = 8.85 × 10⁻¹² F/m. This gives: dΦEdt=35×10-68.85×10-12.
Simplify the expression to calculate the maximum value of dΦ_E/dt. Perform the division of the numerator (35 × 10⁻⁶) by the denominator (8.85 × 10⁻¹²).
The result of the division will give the maximum rate of change of the electric flux (dΦ_E/dt) in units of V·m/s. Ensure that the units are consistent throughout the calculation.
The final value of dΦ_E/dt represents the maximum rate of change of the electric flux between the plates of the capacitor, which is directly proportional to the displacement current.

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Key Concepts

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

Displacement Current

Displacement current is a concept introduced by James Clerk Maxwell to account for changing electric fields in situations where traditional current does not flow, such as in capacitors. It is defined as the rate of change of electric flux through a surface and is crucial for understanding how electric fields behave in capacitive circuits, especially under alternating current (AC) conditions.
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Electric Flux (ΦE)

Electric flux is a measure of the electric field passing through a given area. It is calculated as the product of the electric field strength and the area perpendicular to the field. In the context of capacitors, the electric flux between the plates is essential for determining the displacement current and understanding how the electric field changes over time when an alternating voltage is applied.
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Maxwell's Equations

Maxwell's Equations are a set of four fundamental equations that describe how electric and magnetic fields interact. They encompass the principles of electromagnetism, including how changing electric fields can produce magnetic fields and vice versa. In the context of the problem, these equations help relate the displacement current to the changing electric field between the capacitor plates, allowing for the calculation of the maximum rate of change of electric flux.
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Related Practice
Textbook Question

Suppose that a circular parallel-plate capacitor has radius r₀ = 3.0 cm and plate separation d = 5.0 mm. A sinusoidal potential difference V = V₀ sin (2𝝅ft) is applied across the plates, where V₀ = 180 V and f = 60 Hz. Determine the expression for the amplitude B₀(r) of this time-dependent (sinusoidal) field when r ≤ r₀ and when r > r₀.

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Textbook Question

In an EM wave traveling west, the B field oscillates up and down vertically and has a frequency of 85.0 kHz and an rms strength of 7.75 x 10⁻⁹ T. Determine the frequency and rms strength of the electric field. What is the direction of the electric field oscillations?

Textbook Question

Suppose an air-gap capacitor has circular plates of radius r = 2.5 cm and separation d = 1.6 mm. A 68.0-Hz emf, ε = ε₀ cos ωt, is applied to the capacitor. The maximum displacement current is 35 μA. Determine the maximum conduction current I. Neglect fringing.

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Textbook Question

Suppose that a circular parallel-plate capacitor has radius r₀ = 3.0 cm and plate separation d = 5.0 mm. A sinusoidal potential difference V = V₀ sin (2𝝅ft) is applied across the plates, where V₀ = 180 V and f = 60 Hz. In the region between the plates, show that the magnitude of the induced magnetic field is given by B = B₀(r) cos (2𝝅ft), where B₀(r) is a function of the radial distance r from the capacitor’s central axis.

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Textbook Question

Suppose an air-gap capacitor has circular plates of radius r = 2.5 cm and separation d = 1.6 mm. A 68.0-Hz emf, ε = ε₀ cos ωt, is applied to the capacitor. The maximum displacement current is 35 μA. Determine the value of ε₀. Neglect fringing.