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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 6a
Chapter 30, Problem 6a

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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Understand the problem: The displacement current in a capacitor is equal to the conduction current in the circuit. Therefore, the maximum conduction current I is equal to the maximum displacement current, which is given as 35 μA. This is a key concept in electromagnetism, where the displacement current bridges the gap in the capacitor.
Relate the given quantities to the problem: The emf is given as ε = ε₀ cos(ωt), where ε₀ is the maximum voltage. The angular frequency ω is related to the frequency f by the formula ω = 2πf. Here, f = 68.0 Hz, so calculate ω using ω = 2π × 68.0.
Determine the capacitance of the capacitor: The capacitance C of a parallel-plate capacitor is given by the formula C = (ε₀A) / d, where ε₀ is the permittivity of free space (8.85 × 10⁻¹² F/m), A is the area of the plates (A = πr²), and d is the separation between the plates. Use r = 2.5 cm (convert to meters) and d = 1.6 mm (convert to meters) to calculate C.
Relate the maximum conduction current to the maximum voltage: The current in an AC circuit is related to the voltage and capacitance by the formula I = ωCε₀, where ε₀ is the maximum voltage. Since the maximum conduction current is equal to the maximum displacement current, use the given maximum displacement current (35 μA) to find the relationship between ε₀ and the other quantities.
Solve for the maximum voltage ε₀: Rearrange the formula I = ωCε₀ to solve for ε₀. Substitute the values of I (35 μA), ω (calculated earlier), and C (calculated earlier) to find the maximum voltage ε₀. This completes the solution.

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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 capacitors. It is defined as the rate of change of electric displacement field and is crucial in understanding how capacitors behave in AC circuits. In this scenario, the displacement current is given as 35 μA, which is essential for calculating the conduction current.
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Conduction Current

Conduction current refers to the flow of electric charge through a conductor, typically due to the movement of electrons. In the context of capacitors, the conduction current is related to the displacement current and can be calculated using the relationship between the two. Understanding this relationship is key to solving the problem of determining the maximum conduction current in the given capacitor setup.
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Capacitance and AC Circuits

Capacitance is the ability of a system to store charge per unit voltage and is defined by the formula C = ε₀(A/d), where A is the area of the plates and d is the separation. In AC circuits, the behavior of capacitors changes with frequency, affecting both displacement and conduction currents. Recognizing how capacitance influences these currents is vital for solving the problem presented.
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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. 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 maximum value of dΦE/dt between the plates. Neglect fringing.

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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.