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

Giancoli Douglas 5th edition

Ch. 31 - Maxwell's Equations and Electromagnetic Waves

Problem 6b

Chapter 30, Problem 6b

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.

Verified step by step guidance

Step 1: Understand the relationship between the displacement current and the changing electric field in the capacitor. The displacement current (I_d) is given by the equation:

I d = ε ₀ \cdot A \cdot \frac{d E}{d t}

, where

ε ₀

is the permittivity of free space,

A

is the area of the capacitor plates, and

\frac{d E}{d t}

is the rate of change of the electric field.

Step 2: Express the electric field

E

in terms of the applied emf

ε

. The electric field between the plates is given by

E = \frac{ε}{d}

, where

d

is the separation between the plates. Substituting

ε = ε ₀ cos ω t

, we get

E = \frac{ε}{₀} cos ω t

Step 3: Calculate the rate of change of the electric field

\frac{d E}{d t}

. Differentiating

E = \frac{ε}{₀} cos ω t

with respect to time, we get

\frac{d E}{d t} = \frac{ε}{₀} \cdot ω \cdot sin ω t

. The maximum rate of change occurs when

sin ω t = 1

, so

\frac{d E}{d t} = \frac{ε}{₀} \cdot ω

Step 4: Relate the displacement current to the maximum rate of change of the electric field. Using the formula for displacement current,

I d = ε ₀ \cdot A \cdot \frac{d E}{d t}

, substitute

\frac{d E}{d t} = \frac{ε}{₀} \cdot ω

. This gives

I d = ε ₀ \cdot A \cdot \frac{ε}{₀} \cdot ω

. Rearrange to solve for

ε ₀

ε ₀ = \frac{I}{d}

Step 5: Substitute the known values into the equation. Use

A = π \cdot r^{2}

for the area of the plates,

ω = 2 \cdot π \cdot f

for the angular frequency, and the given values for

I d

r

d

, and

f

. Simplify the expression to find

ε ₀

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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 displacement field (D) and is crucial in understanding how capacitors operate in AC circuits. The displacement current allows for the continuity of current in the presence of a time-varying electric field, which is essential for analyzing capacitive circuits.

Capacitance

Capacitance is the ability of a system to store electric charge per unit voltage. For a parallel plate capacitor, it is given by the formula C = ε₀(A/d), where A is the area of the plates and d is the separation between them. Understanding capacitance is vital for determining how much charge a capacitor can hold and how it responds to an applied voltage, especially in AC circuits where the voltage varies with time.

Maxwell's Equations

Maxwell's Equations are a set of four fundamental equations that describe how electric and magnetic fields interact and propagate. They encompass the laws of electromagnetism, including Gauss's law, Faraday's law of induction, and the concept of displacement current. These equations are essential for understanding the behavior of capacitors in AC circuits, as they govern the relationship between electric fields, magnetic fields, and the currents that flow in response to them.

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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

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

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