Ch. 31 - Maxwell's Equations and Electromagnetic Waves

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

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Giancoli Douglas 5th edition

Ch. 31 - Maxwell's Equations and Electromagnetic Waves

문제 6b

30장, 문제 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.

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.