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 37

Chapter 27, Problem 37

The resistance of a very fine aluminum wire with a 10 μm ×10 μm square cross section is 1000 Ω . A 1000 Ω resistor is made by wrapping this wire in a spiral around a 3.0-mm-diameter glass core. How many turns of wire are needed?

Verified step by step guidance

Determine the resistivity of aluminum (ρ). For aluminum, the resistivity is approximately 2.65 × 10⁻⁸ Ω·m. This value will be used to calculate the length of the wire required to achieve the given resistance.

Use the formula for resistance:

R = \frac{ρ}{A} L

, where R is the resistance (1000 Ω), ρ is the resistivity of aluminum, A is the cross-sectional area of the wire, and L is the length of the wire. Calculate the cross-sectional area A using the given dimensions:

A = 10^{-6} \times 10^{-6} = 10^{-12} m 2

Rearrange the resistance formula to solve for the length of the wire:

L = \frac{R}{\times} ρ

. Substitute the known values: R = 1000 Ω, A = 10⁻¹² m², and ρ = 2.65 × 10⁻⁸ Ω·m. This will give you the total length of the wire required.

Calculate the circumference of the glass core around which the wire is wrapped. The diameter of the core is given as 3.0 mm, so the radius is 1.5 mm or 1.5 × 10⁻³ m. The circumference is given by

C = 2 π r

, where r is the radius. Substitute r = 1.5 × 10⁻³ m to find the circumference.

Divide the total length of the wire (calculated in step 3) by the circumference of the glass core (calculated in step 4) to determine the number of turns of wire needed. This will give you the final answer for the number of turns.

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Key Concepts

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

Resistance

Resistance is a measure of the opposition to the flow of electric current in a conductor. It is determined by the material's properties, length, and cross-sectional area. In this case, the aluminum wire has a specified resistance of 1000 Ω, which is crucial for calculating how many turns of wire are needed to achieve the desired resistance in the resistor.

Ohm's Law

Ohm's Law states that the current (I) through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. This relationship is expressed as V = IR. Understanding this law is essential for analyzing circuits and determining how resistance affects current flow in the wire.

Inductance and Coil Turns

The number of turns in a coil of wire affects its inductance, which is the property that quantifies the coil's ability to store energy in a magnetic field. In this scenario, the wire is wrapped around a glass core, and the total resistance of the coil is influenced by the number of turns. Calculating the required number of turns involves understanding how the geometry of the coil and the wire's resistance contribute to the overall resistance of the resistor.

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