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Chapter 24: Problem 46

How must the diameters of copper and aluminum wire be related if they're to have the same resistance per unit length?

Short Answer

Expert verified
The copper wire diameter must be the square root of the ratio of the resistivities of copper and aluminum, times the diameter of the aluminum wire: \(d_1= \sqrt{\rho_1/\rho_2} * d_2\).

Step by step solution

01

Write out the resistance formula for both wires

Start by writing down the formula for resistance: \(R=\rho L/A\). Remembering that for a circular cross-section \(A=\pi d^2/4\), replace \(A\) with the equivalent dia diameter terms \(A=\pi d^2/4\) in the resistance formula. Write it out separately for each wire. For copper: \(R_1= \rho_1 L/(pi d_1^2/4)\) and for aluminum: \(R_2= \rho_2 L/(pi d_2^2/4)\).
02

Equate the two resistances

Since the resistances of the two wires are equal, their resistances per unit length are also equal, so we can set \(R_1= R_2\). This gives us: \( \rho_1 L/(pi d_1^2/4)= \rho_2 L/(pi d_2^2/4)\). Notice that the terms \(L\) and \(\pi\) are the same on both sides, so we can simplify by cancelling, resulting in the following relationship: \( \rho_1/d_1^2= \rho_2/d_2^2\).
03

Solve for copper wire diameter in terms of aluminum wire diameter

Rearranging terms to find \(d_1\) in terms of \(d_2\), gave: \(d_1 = \sqrt{\rho_1/\rho_2} * d_2\).

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