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

1. Understand that the resistance of a wire is given by the formula: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi> = <mfrac><mi>&#961;</mi><mi>A</mi></mfrac><mi>L</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#961;</mi></math> is the resistivity, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> is the length, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is the cross-sectional area.
2. Recognize that when the wire is stretched to three times its original length, the new length <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L'</mi></math> becomes <math xmlns="http://www.w3.org/1998/Math/MathML">3<mi>L</mi></math>.
3. Since the volume of the wire remains constant, the original volume <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi> = <mi>A</mi><mi>L</mi></math> is equal to the new volume <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V'</mi> = <mi>A'</mi><mi>L'</mi></math>. Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>L</mi> = <mi>A'</mi>3<mi>L</mi></math>.
4. Solve for the new cross-sectional area <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A'</mi></math> using the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A'</mi> = <mfrac><mi>A</mi><mn>3</mn></mfrac></math>.
5. Substitute the new length and area into the resistance formula to find the new resistance <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R'</mi> = <mfrac><mi>&#961;</mi><mi>A'</mi></mfrac><mi>L'</mi></math>, which simplifies to <math xmlns="http://www.w3.org/1998/Math/MathML">9<mi>R</mi></math>.

Key Concepts

Resistance and Resistivity

Resistance (R) is a measure of how much a material opposes the flow of electric current, while resistivity (ρ) is a material property that quantifies how strongly a given material opposes current flow. The resistance of a wire is given by R = ρL/A, where L is the length and A is the cross-sectional area. Understanding how these factors interact is crucial for analyzing changes in resistance when a wire is stretched.

Conservation of Volume

When a ductile wire is stretched, its volume remains constant because the amount of material does not change. This means that if the length increases, the cross-sectional area must decrease proportionally to maintain the same volume. For a wire stretched to three times its original length, the new cross-sectional area will be one-third of the original area, affecting the resistance calculation.

Effect of Length on Resistance

Resistance is directly proportional to the length of the wire. If the length of the wire is increased, the resistance increases because electrons have to travel a longer path, encountering more opposition. In this scenario, stretching the wire to three times its original length increases the resistance by a factor of three, while the cross-sectional area decreases, further affecting the resistance.