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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: 9780137488179Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoGiancoli Douglas 5th editionCh. 31 - Maxwell's Equations and Electromagnetic WavesProblema 6c
Capitolo 30, Problema 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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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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Displacement vs. Distance

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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Electric Flux

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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Kinematics Equations
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