Ch 25: Current, Resistance, and EMF

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 25: Current, Resistance, and EMF

Problem 18

Chapter 25, Problem 18

A ductile metal wire has resistance $R$ . What will be the resistance of this wire in terms of $R$ if it is stretched to three times its original length, assuming that the density and resistivity of the material do not change when the wire is stretched? (Hint: The amount of metal does not change, so stretching out the wire will affect its cross-sectional area.)

Verified step by step guidance

Understand that the resistance of a wire is given by the formula:

R = ρ \times \frac{L}{A}

, where

ρ

is the resistivity,

L

is the length, and

A

is the cross-sectional area.

Recognize that when the wire is stretched to three times its original length, the new length

L'

becomes

3 L

Since the volume of the wire remains constant, the original volume

V = L \times A

is equal to the new volume

V' = L' \times A'

. Therefore,

L \times A = 3 L \times A'

Solve for the new cross-sectional area

A'

A' = \frac{A}{3}

Substitute the new length and area into the resistance formula:

R' = ρ \times \frac{L'}{A'} = ρ \times \frac{3}{1} \times \frac{L}{\frac{A}{3}}

. Simplify to find

R' = 9 R

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Key Concepts

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

Resistance and Resistivity

Resistance (R) of a wire is determined by its resistivity (ρ), length (L), and cross-sectional area (A) using the formula R = ρL/A. Resistivity is a material property that remains constant for a given material, while resistance changes with the wire's dimensions. Understanding this relationship is crucial for analyzing how changes in length and area affect resistance.

Conservation of Volume

When a wire is stretched, its volume remains constant if the material's density does not change. This means that the product of the cross-sectional area (A) and length (L) before stretching equals the product after stretching. This principle helps determine how the cross-sectional area changes when the wire's length is altered.

Effect of Length on Resistance

Stretching a wire increases its length, which directly affects its resistance. Since resistance is proportional to length (R ∝ L), tripling the length of the wire will increase its resistance. However, the cross-sectional area decreases, which further impacts resistance, requiring a comprehensive understanding of how these factors interplay.

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