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Chapter 25: Problem 66

Two cylindrical wires of identical length are made of copper and aluminum. If they carry the same current and have the same potential difference across their length, what is the ratio of their radii?

Short Answer

Expert verified
Answer: The approximate ratio of the radii of the copper and aluminum wires is 1.296.

Step by step solution

01

1. Write the resistance formula for each wire

For the copper wire, the resistance is given by: R_Cu = ρ_Cu * L / A_Cu For the aluminum wire, the resistance is given by: R_Al = ρ_Al * L / A_Al
02

2. Express the area in terms of radii

Both wires are cylindrical, so the cross-sectional area is given by A = πr^2. A_Cu = π * r_Cu^2 A_Al = π * r_Al^2 Thus, we can rewrite the resistance formulas as follows: R_Cu = ρ_Cu * L / (π * r_Cu^2) R_Al = ρ_Al * L / (π * r_Al^2)
03

3. Set the resistances equal to each other

Since both wires have the same current and potential difference, their resistances are equal. We can equate the above formulas: ρ_Cu * L / (π * r_Cu^2) = ρ_Al * L / (π * r_Al^2)
04

4. Simplify and solve for the ratio of radii

We can cancel out the L and π terms from both sides and rearrange the equation to find the ratio of their radii: r_Cu^2 / r_Al^2 = ρ_Al / ρ_Cu Take the square root of both sides: r_Cu / r_Al = sqrt(ρ_Al / ρ_Cu)
05

5. Calculate the ratio using known resistivity values

Now, we can plug in the known resistivity values for copper (ρ_Cu = 1.68 × 10^-8 Ω⋅m) and aluminum (ρ_Al = 2.82 × 10^-8 Ω⋅m) to find the ratio of their radii: r_Cu / r_Al = sqrt((2.82 × 10^-8 Ω⋅m) / (1.68 × 10^-8 Ω⋅m)) r_Cu / r_Al ≈ sqrt(1.679) ≈ 1.296 So, the ratio of the radii of the copper and aluminum wires is approximately 1.296.

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Most popular questions from this chapter

An infinite number of resistors are connected in parallel. If \(R_{1}=10 \Omega, R_{2}=10^{2} \Omega, R_{3}=10^{3} \Omega,\) and so on, show that \(R_{e q}=9 \Omega\).

A copper wire has a diameter \(d_{\mathrm{Cu}}=0.0500 \mathrm{~cm}\) is \(3.00 \mathrm{~m}\) long, and has a density of charge carriers of \(8.50 \cdot 10^{28}\) electrons \(/ \mathrm{m}^{3}\). As shown in the figure, the copper wire is attached to an equal length of aluminum wire with a diameter \(d_{\mathrm{A} \mathrm{I}}=0.0100 \mathrm{~cm}\) and density of charge carriers of \(6.02 \cdot 10^{28}\) electrons \(/ \mathrm{m}^{3}\). A current of 0.400 A flows through the copper wire. a) What is the ratio of the current densities in the two wires, \(J_{\mathrm{Cu}} / J_{\mathrm{Al}} ?\) b) What is the ratio of the drift velocities in the two wires, \(v_{\mathrm{d}-\mathrm{Cu}} / v_{\mathrm{d}-\mathrm{Al}} ?\)

Two conducting wires have identical lengths \(L_{1}=L_{2}=\) \(L=10.0 \mathrm{~km}\) and identical circular cross sections of radius \(r_{1}=r_{2}=r=1.00 \mathrm{~mm} .\) One wire is made of steel (with resistivity \(\rho_{\text {steel }}=40.0 \cdot 10^{-8} \Omega \mathrm{m}\) ); the other is made of copper (with resistivity \(\left.\rho_{\text {copper }}=1.68 \cdot 10^{-8} \Omega \mathrm{m}\right)\) a) Calculate the ratio of the power dissipated by the two wires, \(P_{\text {copper }} / P_{\text {steel }},\) when they are connected in parallel; a potential difference of \(V=100 . \mathrm{V}\) is applied to them. b) Based on this result, how do you explain the fact that conductors for power transmission are made of copper and not steel?

A modern house is wired for \(115 \mathrm{~V}\), and the current is limited by circuit breakers to a maximum of \(200 .\) A. (For the purpose of this problem, treat these as DC quantities.) a) Calculate the minimum total resistance the circuitry in the house can have at any time. b) Calculate the maximum electrical power the house can consume.

The Stanford Linear Accelerator accelerated a beam consisting of \(2.0 \cdot 10^{14}\) electrons per second through a potential difference of \(2.0 \cdot 10^{10} \mathrm{~V}\) a) Calculate the current in the beam. b) Calculate the power of the beam. c) Calculate the effective ohmic resistance of the accelerator.
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