A length of # 8 copper wire (radius =1.63 mm ) has a mass of 24.0 kg and a resistance of 2.061 o

step 1 All right, the problem 131. This question, we're going to calculate the overall resistance for t

A length of # 8 copper wire (radius =1.63 mm ) has a mass of...

Key Concepts

Mass, Density, and Geometric Relationships

The relationship between mass, density, and volume is used to deduce the geometric properties of a material, such as its length when the mass and cross-sectional area (or diameter) are known. This concept is important in practical applications where the physical dimensions of a conductor must be determined from its measured mass and known material density.

Unit Conversion and Dimensional Analysis

Accurate electrical calculations require careful unit conversion and dimensional consistency. Whether converting lengths, areas, or resistance values provided per unit length, maintaining consistency in units is essential to obtaining correct and meaningful results.

Electrical Resistance

Electrical resistance quantifies how much a material opposes the flow of electric current. It depends on factors such as the material’s intrinsic resistivity, the length of the conductor, and its cross-sectional area. The formula R = ?L/A, where ? is the resistivity, L is the length, and A is the cross-sectional area, is fundamental in relating these properties.

Resistivity

Resistivity is an inherent property of a material that indicates its ability to resist electric current. It varies from one material to another and is crucial when calculating resistance because it directly affects how much resistance a conductor will have over a given length and cross-sectional area.

Cross-sectional Area and Geometry

The geometry of a conductor, particularly its cross-sectional area, plays a critical role in determining its resistance. A larger cross-sectional area allows more electrons to pass through simultaneously, thereby reducing resistance, while a longer conductor increases resistance according to the R = ?L/A relationship.

Transcript

00:01 All right, the problem 131.

00:03 This question, we're going to calculate the overall resistance for the copper wire.

00:13 So first, we know the relationship between density, mass, and the volume of copper wire is given by the equation.

00:20 Density equal to mass, divided by the volume.

00:28 So by rearranging this equation, we can have the following equation.

00:33 Volume equal to mass divided by the density.

00:40 So the density is given by the problem.

00:45 Density of the copper is equal to 8 .96 times grams per centimeter cubed.

00:54 And the mass from the problem we know that is 24 kilograms.

01:02 And we know that 1 kilogram is equal to 10 to the 3rd.

01:07 Grams.

01:10 So converting this mass to gram, we have 24 times 10 to the third gram per one kilogram and that is 24100 grams.

01:28 So now we have the grams of the copper and density of the copper by substituting this values into the equation of the volume.

01:39 We have the v equal to m divided by d that is equal to 2 4000 grams divided by 8 .96 gram per centimeter cube and we get the volume equal to 2679 centimeter cube so here this is the volume of copper water.

02:18 All right then we also know the relationship between volume radius and length of a copper wire is given by v equal to pi r square l...

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