Ch 27: Current and Resistance

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 27: Current and Resistance

Problem 62

Chapter 27, Problem 62

What diameter should the nichrome wire in FIGURE P27.62 be in order for the electric field strength to be the same in both wires?

Verified step by step guidance

Step 1: Understand the problem. The goal is to find the diameter of the nichrome wire such that the electric field strength in both wires is the same. Electric field strength in a wire is given by E = ρ * J, where ρ is the resistivity of the material and J is the current density.

Step 2: Relate the current density to the wire's cross-sectional area. Current density J is defined as J = I / A, where I is the current and A is the cross-sectional area of the wire. For a cylindrical wire, A = π * (d/2)^2, where d is the diameter of the wire.

Step 3: Set up the condition for equal electric field strength. Since E = ρ * J, and we want the electric field strength to be the same in both wires, we can write ρ_aluminum * J_aluminum = ρ_nichrome * J_nichrome.

Step 4: Substitute the expressions for current density into the equation. Using J = I / A, the equation becomes ρ_aluminum * (I / A_aluminum) = ρ_nichrome * (I / A_nichrome). Cancel out the current I, as it is the same in both wires.

Step 5: Solve for the diameter of the nichrome wire. Substitute A_aluminum = π * (d_aluminum/2)^2 and A_nichrome = π * (d_nichrome/2)^2 into the equation. Use the given diameter of the aluminum wire (1.0 mm) and the resistivities of aluminum and nichrome to find the diameter of the nichrome wire.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electric Field Strength

Electric field strength is defined as the force per unit charge experienced by a positive test charge placed in the field. It is a vector quantity, represented by the symbol E, and is measured in volts per meter (V/m). Understanding electric field strength is crucial for analyzing how different materials respond to electric currents, particularly in comparing wires of different materials and diameters.

Resistivity and Conductivity

Resistivity is a material property that quantifies how strongly a given material opposes the flow of electric current. It is denoted by the symbol ρ (rho) and is measured in ohm-meters (Ω·m). Conductivity, the inverse of resistivity, indicates how easily electricity can flow through a material. The relationship between resistivity, electric field strength, and current is essential for determining the appropriate diameter of the nichrome wire to match the electric field strength of the aluminum wire.

Current Density

Current density is defined as the amount of electric current flowing per unit area of a cross-section of a conductor. It is represented by the symbol J and is measured in amperes per square meter (A/m²). The current density is directly related to the electric field strength and the material's conductivity, which means that to achieve the same electric field strength in both wires, the current density must be equal, necessitating a specific diameter for the nichrome wire.

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Related Practice

Textbook Question

The total amount of charge in coulombs that has entered a wire at time t is given by the expression Q=4t−t², where t is in seconds and t≥0. Graph I versus t for the interval 0≤t≤4 s.

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Textbook Question

An aluminum wire consists of the three segments shown in FIGURE P27.64. The current in the top segment is 10 A. For each of these three segments, find the current density J. Place your results in a table for easy viewing.

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Textbook Question

The current supplied by a battery slowly decreases as the battery runs down. Suppose that the current as a function of time is $I = (0.75 A) e^{- \frac{t}{6 h}}$ . What is the total number of electrons transported from the positive electrode to the negative electrode by the charge escalator from the time the battery is first used until it is completely dead?

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Textbook Question

The total amount of charge that has entered a wire at time t is given by the expression $Q = (20 C) (1 - e^{- \frac{t}{2.0 s}})$ , where t is in seconds and t≥0. What is the maximum value of the current?

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Textbook Question

An aluminum wire consists of the three segments shown in FIGURE P27.64. The current in the top segment is 10 A. For each of these three segments, find the drift velocity v_d. Place your results in a table for easy viewing.

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Textbook Question

The two wires in FIGURE P27.63 are made of the same material. What is the electron drift speed in the 2.0-mm-diameter segment of the wire?

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