Ch 28: Fundamentals of Circuits

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 28: Fundamentals of Circuits

Problem 47b

Chapter 28, Problem 47b

A small toaster that operates at 120 V has a heating element made from a 4.4-m-long, 0.70-mm-diameter nichrome wire. The resistivity, density, and specific heat of nichrome are, respectively, 1.5 x 10⁻⁶ Ωm, 8400 kg/m³, and 450 J/kg K. If half the heat energy is lost to the air, how long does it take the heating element to warm from 20℃ to 450℃, about the temperature at which it first begins to glow red?

Verified step by step guidance

Determine the resistance of the nichrome wire using the formula for resistance:

R = \frac{ρ}{L} .

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Key Concepts

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

Ohm's Law

Ohm's Law states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance. This relationship is expressed as V = IR, where V is voltage, I is current, and R is resistance. Understanding this law is crucial for analyzing electrical circuits, such as the toaster's heating element, to determine how much current flows at a given voltage.

Heat Transfer and Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius. In this context, it helps determine how much energy is needed to heat the nichrome wire from 20℃ to 450℃. Additionally, understanding heat transfer mechanisms, including conduction, convection, and radiation, is essential for calculating energy losses to the surrounding air, which affects the heating time.

Resistivity and Resistance

Resistivity is a material property that quantifies how strongly a given material opposes the flow of electric current, expressed in ohm-meters (Ωm). The resistance of a wire can be calculated using the formula R = ρ(L/A), where ρ is resistivity, L is the length of the wire, and A is its cross-sectional area. This concept is vital for determining how much electrical energy is converted into heat in the toaster's heating element, influencing its performance and efficiency.

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Related Practice

Textbook Question

A lightbulb is in series with a 2.0 Ω resistor. The lightbulb dissipates 10 W when this series circuit is connected to a 9.0 V battery. What is the current through the lightbulb? There are two possible answers; give both of them.

Textbook Question

Suppose you have resistors 2.5 Ω, 3.5 Ω, and 4.5 Ω and a 100 V power supply. What is the ratio of the total power delivered to the resistors if they are connected in parallel to the total power delivered if they are connected in series?

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Textbook Question

Load resistor R is attached to a battery of emf and internal resistance r. For what value of the resistance R, in terms of ∈ and r, will the power dissipated by the load resistor be a maximum?

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Textbook Question

A 2.0-m-long, 1.0-mm-diameter wire has a variable resistivity given by where x is measured from one end of the wire. What is the current if this wire is connected to the terminals of a 9.0 V battery?

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Textbook Question

You have a 2.0 Ω resistor, a 3.0 Ω resistor, a 6.0 Ω resistor, and a 6.0 V battery. Draw a diagram of a circuit in which all three resistors are used and the battery delivers 9.0 W of power.

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Textbook Question

To which two points in the circuit of FIGURE P28.45 should a 12 V battery be connected to dissipate the most power?

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