Ch. 25 - Electric Current and Resistance

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179당신이 사용하는 게 아니라요?교과서 변경

Giancoli Douglas 5th edition

Ch. 25 - Electric Current and Resistance

문제 95c

24장, 문제 95c

Household wiring has sometimes used aluminium instead of copper. What would be the resistance of the same wire if it were made of aluminum?

검증된 단계별 안내

Step 1: Understand the problem. The goal is to calculate the resistance of a wire made of aluminum, given that the wire has the same dimensions as a copper wire. Resistance depends on the material's resistivity, the length of the wire, and its cross-sectional area.

Step 2: Recall the formula for resistance: \( R = \frac{\rho \cdot L}{A} \), where \( R \) is resistance, \( \rho \) is resistivity, \( L \) is the length of the wire, and \( A \) is the cross-sectional area. The resistivity of aluminum is different from copper, so you will need the resistivity value for aluminum.

Step 3: Identify the values provided or implied in the problem. The length \( L \) and cross-sectional area \( A \) of the wire are the same as those of the copper wire. Look up the resistivity of aluminum (typically \( \rho_{Al} \approx 2.82 \times 10^{-8} \, \Omega \cdot \text{m} \)).

Step 4: Substitute the resistivity of aluminum \( \rho_{Al} \), the length \( L \), and the cross-sectional area \( A \) into the formula \( R = \frac{\rho \cdot L}{A} \). Ensure the units are consistent (e.g., meters for length and square meters for area).

Step 5: Simplify the expression to find the resistance of the aluminum wire. While you won't calculate the final value here, ensure you understand how the resistivity of aluminum affects the resistance compared to copper.

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주요 개념

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Electrical Resistance

Electrical 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. The resistance (R) can be calculated using Ohm's Law, R = V/I, where V is voltage and I is current. Different materials have different resistivities, which significantly affect their resistance.

Resistivity

Resistivity is a fundamental property of materials that quantifies how strongly they resist the flow of electric current. It is denoted by the symbol ρ (rho) and is measured in ohm-meters (Ω·m). For example, copper has a lower resistivity than aluminum, meaning it conducts electricity more efficiently. The resistivity of a material is temperature-dependent and varies with its physical state.

Material Comparison

When comparing materials like aluminum and copper for electrical wiring, it's essential to consider their conductivity, cost, weight, and resistance. While copper has better conductivity and lower resistance, aluminum is lighter and often cheaper. However, aluminum's higher resistance can lead to greater energy losses in electrical systems, making it crucial to calculate the resistance accurately when substituting one for the other.

관련 실천

교과서 질문

Small changes in the length of an object can be measured using a strain gauge sensor, which is a wire that when undeformed has length ℓ₀, cross-sectional area A₀, and resistance R₀. This sensor is rigidly affixed to the object’s surface, aligning its length in the direction in which length changes are to be measured. As the object deforms, the length of the wire sensor changes by Δℓ, and the resulting change ΔR in the sensor’s resistance is measured. Assuming that as the solid wire is deformed to a length ℓ, its density and volume remain constant (only approximately valid), show that the strain ( = Δℓ / ℓ₀ ) of the wire sensor, and thus of the object to which it is attached, is approximately ΔR / 2R₀.

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교과서 질문

The level of liquid helium (temperature ≈ 4K) in its storage tank can be monitored using a vertically aligned niobium–titanium (NbTi) wire, whose length ℓ spans the height of the tank. In this level-sensing setup, an electronic circuit maintains a constant electrical current I at all times in the NbTi wire and a voltmeter monitors the voltage V across this wire. The NbTi wire is superconducting ( R = 0) if below its transition temperature of 10 K, so the portion of the wire immersed in the liquid helium is in the superconducting state, while the portion above the liquid (in helium vapor with temperature above 10 K) is in the normal state. Define ƒ = x/ℓ to be the fraction of the tank filled with liquid helium (Fig. 25–40) and V₀ to be the value of the voltage V when the tank is empty (ƒ = 0) . Determine the relation between f and V (in terms of V₀).

763

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교과서 질문

Household wiring has sometimes used aluminium instead of copper.Typical copper wire used for home wiring in the U.S. has a diameter of 1.63 mm. What is the resistance of 125 m of this wire?

1136

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교과서 질문

Estimate how far can an average electron move along one of the connecting wires of a 750-W toaster during an alternating current cycle? The power cord has copper wires of diameter 1.7 mm and is plugged into a 60-Hz 120-V ac outlet. [Hint: For sinusoidal motion, Chapter 14, we saw that the maximum distance traveled from equilibrium (amplitude A) is proportional to the maximum (drift) speed (Eq. 14–9a). This maximum drift speed is related to the maximum current (Section 25–8), which is calculated as the first step here; see Chapter 14.]

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교과서 질문

Copper wire of diameter 0.259 cm is used to connect a set of appliances at 120 V, which draw 1250 W of power total. What power is wasted in 25.0 m of this wire?

1103

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