Ch 28: Fundamentals of Circuits

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 28: Fundamentals of Circuits

Problem 13

Chapter 28, Problem 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?

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Related Practice

Textbook Question

Draw a graph of the potential as a function of the distance traveled through the circuit, traveling cw from V = 0 V at the lower left corner.

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Textbook Question

A typical American family uses 1000 kWh of electricity a month. What is the average current in the 120 V power line to the house?

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Textbook Question

Many electric companies use time-of-day pricing in which electricity costs more during hours of high demand. Suppose electricity costs \(0.21/kWh from 10 a.m. to 6 p.m.; \)0.09/kWh at all other times. What is the annual cost of electricity for a 2.5 kW industrial pump that runs 24 hours a day?

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Textbook Question

The five identical bulbs in FIGURE EX28.11 are all glowing. The battery is ideal. What is the order of brightness of the bulbs, from brightest to dimmest? Some may be equal.

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Textbook Question

A 60 W lightbulb and a 100 W lightbulb are placed in the circuit shown in FIGURE EX28.9. Both bulbs are glowing. Which bulb is brighter? Or are they equally bright?

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Textbook Question

A typical American family uses 1000 kWh of electricity a month. On average, what is the resistance of a household?

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