A 1.50-m cylindrical rod of diameter 0.500 cm is connected...

A 1.50-m cylindrical rod of diameter 0.500 cm is connected to a power supply that maintains a constant potential difference of 15.0 V across its ends, while an ammeter measures the current through it. You observe that at room temperature (20.0$^\circ$C) the ammeter reads 18.5 A, while at 92.0$^\circ$C it reads 17.2 A. You can ignore any thermal expansion of the rod. Find (a) the resistivity at 20.0$^\circ$C and (b) the temperature coefficient of resistivity at 20$^\circ$C for the material of the rod.

Key Concepts

Ohm's Law

Ohm's Law states that the voltage across a conductor is directly proportional to the current through it, with the constant of proportionality being the resistance. This fundamental relationship, expressed as V = IR, is crucial for analyzing electrical circuits and understanding how changes in resistance affect current when a constant voltage is applied.

Resistance and Resistivity

Resistance is an intrinsic property of a material that quantifies how strongly it opposes the flow of electric current. The resistance of a conductor is related to its resistivity, length, and cross-sectional area by the formula R = ?L/A. This relationship is essential in predicting how changes in physical dimensions or material properties influence the overall resistance.

Geometric Considerations in Electrical Conduction

The geometry of a conductor, specifically its length and cross-sectional area, plays a critical role in determining its resistance. A longer conductor increases the resistance, while a larger cross-sectional area decreases it. Calculating the cross-sectional area from diameter or radius measurements is a common step in applying the resistance formula R = ?L/A.

Temperature Dependence of Resistivity

The resistivity of most materials changes with temperature. Typically, for conductors, resistivity increases as temperature increases. This temperature dependence can often be approximated as a linear relationship for small temperature ranges, which is key to understanding and predicting material behavior in varying thermal environments.

Temperature Coefficient of Resistivity

The temperature coefficient of resistivity is a parameter that quantifies how the resistivity of a material changes with temperature. It is defined in the linear approximation of resistivity variation as ? = ?0[1 + ?(T - T0)], where ? represents the temperature coefficient. This concept is important for designing and analyzing electrical systems that operate over a range of temperatures.

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