A ductile metal wire has resistance R. What will be the...

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.)

Key Concepts

Application of Proportional Reasoning

By applying proportional reasoning to the relationship between a conductor’s geometry and its resistance, one can determine how changes in length and cross-sectional area influence the resistance. This involves comparing the scaling factors for length and area and using them to adjust the resistance according to the formula R = ?(L/A).

Resistivity and Ohm's Law

Resistance in an electrical conductor is determined by the material's resistivity through the relation R = ?(L/A), where ? is the resistivity, L is the length, and A is the cross-sectional area. This relationship, known as Ohm's Law in physical form, links how the innate material property (resistivity) and the conductor’s geometry determine the opposition to the flow of electric current.

Relationship Between Geometry and Resistance

The resistance of a conductor is directly proportional to its length and inversely proportional to its cross-sectional area. This means that if a conductor's dimensions are altered—for example, increasing its length while reducing its area—the resistance changes in a predictable manner according to the formula R = ?L/A.

Conservation of Volume

When a material is deformed (as when a wire is stretched), the total volume of the material remains constant if no material is added or removed. This constraint implies that any increase in length must be accompanied by a corresponding decrease in cross-sectional area, preserving the overall volume. Understanding this principle is crucial for calculating how changes in shape affect properties like resistance.

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