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Chapter 24: Problem 39

A copper wire joins an aluminum wire whose diameter is twice that of the copper. The same current flows in both wires. The density of conduction electrons in copper is \(1.1 \times 10^{29} \mathrm{m}^{-3}\); in aluminum it's \(2.1 \times 10^{29} \mathrm{m}^{-3} .\) Compare (a) the drift speeds and (b) the current densities in each.

Short Answer

Expert verified
The drift speed in the aluminum wire is approximately 7.6 times less than that of the copper wire, and the current density in the aluminum wire is 4 times less than in the copper wire.

Step by step solution

01

Identifying Drift Speed

Drift speed of electrons \(v_d\) can be calculated using the formula \(v_d = I / (n \cdot A \cdot e)\) where \(I\) is the current, \(n\) is the density of conduction electrons, \(A\) is the cross section area of the wire and \(e\) approximates 1.6 x 10^-19 coulombs, which is the charge of an electron. As both wires have the same current, the drift speed is inversely proportional to the product of density of conduction electrons \(n\) and the cross section area \(A\). It's also given that the diameter of the aluminum wire is twice that of copper, so the area of aluminum will be four times (as area \(A = \pi \cdot (d/2)^2\)) that of copper.
02

Calculating Drift Speed

Considering the Area and density, we can say that the drift speed of Aluminum wire will be \(v_{d,Al} = v_{d, Cu} / (4 \cdot (2.1/1.1)) = v_{d, Cu} / 7.6\). This shows that the drift speed in the aluminum wire will be approximately 7.6 times less than in the copper wire.
03

Identifying Current Density

Current density \(J\) can be defined as the current \(I\) flowing per cross-sectional area \(A\) and can be calculated using the formula \(J = I / A\). Since both wires conduct the same current, the current density of the wires is inversely proportional to the cross-sectional area. Since, the aluminum wire has twice the diameter of the copper wire, the cross sectional area for aluminum will be four times as much as that for the copper wire.
04

Calculating Current Density

Considering the areas, we can say that the current density of the Aluminum wire will be \(J_{Al} = J_{Cu} / 4\). This shows that the current density in the aluminum wire will be 4 times less than the copper wire.

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