You want to produce three 1.00 -mm-diameter cylindrical...

You want to produce three 1.00 -mm-diameter cylindrical wires, each with a resistance of $1.00 \Omega$ at room temperature. One wire is gold, one is copper, and one is aluminum. Refer to Table 25.1 for the resistivity values. (a) What will be the length of each wire? (b) Gold has a density of $1.93 \times 10^{4} \mathrm{~kg} / \mathrm{m}^{3}$. What will be the mass of the gold wire? If you consider the current price of gold, is this wire very expensive?

You want to produce three 1.00 -mm-diameter cylindrical wires, each with a resistance of $1.00 \Omega$ at room temperature. One wire is gold, one is copper, and one is aluminum. Refer to Table 25.1 for the resistivity values. (a) What will be the length of each wire?
(b) Gold has a density of $1.93 \times 10^{4} \mathrm{~kg} / \mathrm{m}^{3}$. What will be the mass of the gold wire? If you consider the current price of gold, is this wire very expensive?

Key Concepts

Mass Determination from Density

The mass of an object can be calculated by multiplying its volume by its density. In the case of a cylindrical wire, once the length and cross-sectional area are known, the volume is the product of these two, and multiplying by the material’s density yields the mass.

Cost Analysis of Materials

In addition to physical parameters, the practical feasibility of using a material often involves cost analysis. This includes evaluating the amount of material required to achieve desired electrical properties and its corresponding market price to determine if the design is economically viable.

Cylindrical Geometry and Cross-sectional Area

Understanding the geometry of a cylindrical wire is crucial because its resistance depends on its cross-sectional area. The area, typically calculated using the diameter or radius of the cylinder, directly influences the resistance since a larger area allows more current to pass with less opposition.

Resistivity

Resistivity is a fundamental property of a material that quantifies how strongly it opposes the flow of electric current. It is used in calculating resistance and is dependent on the material, temperature, and sometimes the structure or purity of the substance.

Resistance Calculation in Conductors

The resistance of a conductor can be determined using the formula R = ?L/A, where R is the resistance, ? is the resistivity, L is the length, and A is the cross-sectional area. This relationship is fundamental in designing circuits and determining the necessary dimensions to achieve a desired resistance.

Transcript

00:01 To solve this problem.

00:01 We identified the things that were given and we're given the diameter of the wire's have, which is one millimeter.

00:10 Her tens of meters, three meters.

00:12 There resistive ity turned a resistance.

00:15 Rather, one home the materials were asked to work with is a gold, copper and aluminum, and the resistive ity can be found in table twenty five point one, and in part a of the problem.

00:31 We were asked to determine the length of each wire that will give us a resistance of one for a diameter of one millimeter of each of the wires.

00:44 So in order to solve this problem her to solve part ay, we use equation twenty five ten.

00:55 But if i find one home and this equation tells us that the resistance is equal to resistive ity times the length of the wire divided by the cross sectional area in here we can just rearrange this equation so that we can isolate the length so the length is equal to the resistance.

01:26 Time's a cross sectional area divided by resistive ity.

01:32 So we're given the resistance.

01:37 We're given the resistive ity in table twenty five foot one, and we can easily calculate the cross sectional area are remembering that the wire is a cylinder and a circle for cross sectional area.

01:50 So that's just pi r squared, which is high times one half the diameter squared to know we have all the rather than pieces of information to find the length of each of the wires that were asked to work with.

02:12 I'll do this i'll calculate the actual numbers on the next page.

02:20 So the length ah apologised we read the equation here, something over work with.

02:26 So the length is the resistance have the cross sectional area divided by the resistive ity.

02:35 So this is our main crazy here.

02:40 So like the gold wire that we need no one home times the cross sectional area which is one half times one times ten today minus three meter square times pi.

03:06 So this portion is the cross sectional area divided by the resistive ity of gold, which is two point four one four four times ten to the minus eight bold meter and you'll notice that this equation gives you the right dimensions for the length which is meters is the holmes cancel out and a meter from the denominator.

03:40 Cancel out one of the meters from the numerator in your lap with just meters, so lengthen cold after you run these numbers so that it is thirty two point one seven meters.

04:04 In a similar way, we compute the length of the copper and the aluminum wire...

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