Ch 28: Fundamentals of Circuits

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 28: Fundamentals of Circuits

문제 13

28장, 문제 13

A 100 W (120 V) incandescent lightbulb contains a 7.0-cm-long tungsten filament. The high-temperature resistivity of tungsten is 9.0 x 10⁻⁷ Ωm. What is the diameter of the filament?

검증된 단계별 안내

Step 1: Start by recalling the formula for electrical power:

P = V^{2} / R

, where

P

is power,

V

is voltage, and

R

is resistance. Rearrange this formula to solve for resistance:

R = V^{2} / P

Step 2: Substitute the given values for voltage (

V = 120

V) and power (

P = 100

W) into the formula to calculate the resistance of the filament:

R = 120^{2} / 100

Step 3: Use the formula for resistance in terms of resistivity:

R = ρ L / A

, where

ρ

is the resistivity,

L

is the length of the filament, and

A

is the cross-sectional area. Rearrange this formula to solve for the cross-sectional area:

A = ρ L / R

Step 4: Substitute the given values for resistivity (

ρ = 9.0 × 10

^-7 Ωm), length (

L = 0.07

m), and the resistance calculated in Step 2 into the formula for

A

to find the cross-sectional area.

Step 5: Recall that the cross-sectional area of a cylindrical filament is given by

A = π (d / 2)

², where

d

is the diameter. Rearrange this formula to solve for the diameter:

d = 2 √(A / π)

. Substitute the value of

A

from Step 4 to calculate the diameter of the filament.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

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

Resistivity

Resistivity is a material property that quantifies how strongly a given material opposes the flow of electric current. It is denoted by the symbol 'ρ' and is measured in ohm-meters (Ωm). For tungsten, the high-temperature resistivity is given as 9.0 x 10⁻⁷ Ωm, which is crucial for calculating the filament's resistance and, subsequently, its diameter.

Resistance

Resistance is the opposition to the flow of electric current in a conductor, measured in ohms (Ω). It can be calculated using the formula R = ρ(L/A), where R is resistance, ρ is resistivity, L is the length of the conductor, and A is the cross-sectional area. Understanding resistance is essential for determining the filament's diameter based on its length and resistivity.

Cross-sectional Area

The cross-sectional area (A) of a cylindrical object, like a filament, is the area of a slice taken perpendicular to its length. For a circular filament, A can be calculated using the formula A = π(d/2)², where d is the diameter. This area is vital for relating the filament's resistance to its physical dimensions, allowing us to find the diameter from the known resistance and length.