Ch 26: Potential and Field

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 26: Potential and Field

문제 21a

26장, 문제 21a

Two 3.0-cm-diameter aluminum electrodes are spaced 0.50 mm apart. The electrodes are connected to a 100 V battery. What is the capacitance?

검증된 단계별 안내

Step 1: Understand the formula for capacitance of a parallel plate capacitor. The capacitance is given by the formula:

C = (ε A) / d

, where

ε

is the permittivity of the material between the plates,

A

is the area of the plates, and

d

is the separation between the plates.

Step 2: Identify the values given in the problem. The diameter of the electrodes is 3.0 cm, so the radius is 1.5 cm or 0.015 m. The separation between the plates is 0.50 mm or 0.0005 m. The material between the plates is air, so the permittivity is approximately

ε = 8.85 \times 10 ⁻ 12 F / m

Step 3: Calculate the area of the circular plates. The area of a circle is given by

A = π r ²

. Substitute the radius

r = 0.015 m

into the formula to find the area.

Step 4: Substitute the values for

ε

A

, and

d

into the capacitance formula. Use

ε = 8.85 \times 10 ⁻ 12

, the calculated area, and

d = 0.0005 m

Step 5: Simplify the expression to find the capacitance. Ensure the units are consistent throughout the calculation, and express the final capacitance in Farads (F).

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주요 개념

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

Capacitance

Capacitance is the ability of a system to store electric charge per unit voltage. It is defined as the ratio of the electric charge (Q) stored on the electrodes to the potential difference (V) across them, expressed as C = Q/V. The unit of capacitance is the farad (F), and it is influenced by the physical characteristics of the capacitor, including the area of the electrodes and the distance between them.

Parallel Plate Capacitor

A parallel plate capacitor consists of two conductive plates separated by an insulating material (dielectric). The capacitance of such a capacitor can be calculated using the formula C = ε₀(A/d), where ε₀ is the permittivity of free space, A is the area of one of the plates, and d is the separation between the plates. This model simplifies the analysis of capacitors and is fundamental in understanding how they operate.

Electric Field

The electric field (E) is a vector field that represents the force per unit charge experienced by a positive test charge placed in the field. In the context of capacitors, the electric field between the plates is uniform and can be calculated using E = V/d, where V is the voltage across the plates and d is the separation distance. Understanding the electric field is crucial for analyzing how capacitors store energy and influence charge distribution.