Log out Go to App
Go to App

Learning Materials

Features

Discover

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A battery has a potential difference of \(14.50 \mathrm{~V}\) when it is not connected in a circuit. When a \(17.91-\Omega\) resistor is connected across the battery, the potential difference of the battery drops to \(12.68 \mathrm{~V}\). What is the internal resistance of the battery?

Two resistors with resistances \(200 . \Omega\) and \(400 . \Omega\) are connected (a) in series and (b) in parallel with an ideal 9.00-V battery. Compare the power delivered to the \(200 .-\Omega\) resistor.

A material is said to be ohmic if an electric field, \(\vec{E}\), in the material gives rise to current density \(\vec{J}=\sigma \vec{E},\) where the conductivity, \(\sigma\), is a constant independent of \(\vec{E}\) or \(\vec{J}\). (This is the precise form of Ohm's Law.) Suppose in some material an electric field, \(\vec{E}\), produces current density, \(\vec{J},\) not necessarily related by Ohm's Law; that is, the material may or may not be ohmic. a) Calculate the rate of energy dissipation (sometimes called ohmic heating or joule heating) per unit volume in this material, in terms of \(\vec{E}\) and \(\vec{J}\). b) Express the result of part (a) in terms of \(\vec{E}\) alone and \(\vec{J}\) alone, for \(\vec{E}\) and \(\vec{J}\) related via Ohm's Law, that is, in an ohmic material with conductivity \(\sigma\) or resistivity \(\rho .\)

A voltage spike causes the line voltage in a home to jump rapidly from \(110 . \mathrm{V}\) to \(150 . \mathrm{V}\). What is the percentage increase in the power output of a 100.-W tungsten-filament incandescent light bulb during this spike, assuming that the bulb's resistance remains constant?

What is the current density in an aluminum wire having a radius of \(1.00 \mathrm{~mm}\) and carrying a current of \(1.00 \mathrm{~mA}\) ? What is the drift speed of the electrons carrying this current? The density of aluminum is \(2.70 \cdot 10^{3} \mathrm{~kg} / \mathrm{m}^{3},\) and 1 mole of aluminum has a mass of \(26.98 \mathrm{~g}\). There is one conduction electron per atom in aluminum.
See all solutions

Recommended explanations on Physics Textbooks

Physical Quantities And Units

Read Explanation

Dynamics

Read Explanation

Quantum Physics

Read Explanation

Radiation

Read Explanation

Translational Dynamics

Read Explanation

Particle Model of Matter

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free