Two cylindrical resistors are made from the same material...

Name: Problem 63, Chapter 24: Essential University Physics Global Edition
Uploaded: 2020-05-09T20:45:43-08:00
Duration: 1 min 33 s
Channel: Mrinal Rana
Description: Two cylindrical resistors are made from the same material and have the same length. When connected across the same battery, one dissipates twice as much power as the other. How do their diameters compare?

Two cylindrical resistors are made from the same material and have the same length. When connected across the same battery, one dissipates twice as much power as the other. How do their diameters compare?

Key Concepts

Electrical Resistance and Geometry

This concept describes how the resistance of a material depends on its physical dimensions and material properties. For a uniform cylindrical resistor, resistance is calculated as R = ?L/A, where ? is resistivity, L is the length, and A is the cross-sectional area. Since the area for a cylinder is given by A = (?d²)/4, the resistance is inversely proportional to the square of the diameter, demonstrating how changes in geometry directly affect electrical resistance.

Power Dissipation in Resistive Elements

This concept explains how electrical power is converted into heat in a resistor. With a constant voltage supply, the power dissipated can be expressed as P = V²/R, meaning that a resistor with a lower resistance will dissipate more power. Understanding this relationship is essential when analyzing how variations in resistor dimensions, and consequently their resistance, influence the power they dissipate.

Scaling Laws in Physics

Scaling laws explain how physical quantities change when the dimensions of a system are altered. In the context of cylindrical resistors, the key idea is that the cross-sectional area, and therefore the resistance, scales as the square of the diameter. This quadratic relationship means that even a small change in diameter can significantly affect the resistance and the resultant power dissipation in the resistor.

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