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Ch 27: Current and Resistance
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 27: Current and ResistanceProblema 62
Capitolo 27, Problema 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?
Illustration of nichrome and aluminum wires, highlighting the aluminum wire's 1.0 mm diameter for electric field comparison.

Guida verificata passo dopo passo
1
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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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.
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Intro to Electric Fields

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.
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Conduction

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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Intro to Density
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