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27. Resistors & DC Circuits

Power in Circuits

Physics

27. Resistors & DC Circuits

Power in Circuits

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4:26 minutes

Problem 45

Textbook Question

A 1.0-m-long round tungsten wire is to reach a temperature of 3100 K when a current of 18.0 A flows through it. What diameter should the wire be? Assume the wire loses energy only by radiation (emissivity ∊ = 1.0, Section 19–10) and the surrounding temperature is 20°C.

Verified step by step guidance

Start by identifying the key concepts involved in the problem. The wire loses energy by radiation, so we will use the Stefan-Boltzmann law for radiative heat loss: $ P = \sigma \epsilon A (T^4 - T_s^4) $, where $ P $ is the power radiated, $ \sigma $ is the Stefan-Boltzmann constant, $ \epsilon $ is the emissivity, $ A $ is the surface area, $ T $ is the wire's temperature, and $ T_s $ is the surrounding temperature.

The power $ P $ is also related to the electrical energy dissipated in the wire due to resistance: $ P = I^2 R $, where $ I $ is the current and $ R $ is the resistance of the wire. The resistance $ R $ can be expressed as $ R = \rho \frac{L}{A_c} $, where $ \rho $ is the resistivity of tungsten, $ L $ is the length of the wire, and $ A_c $ is the cross-sectional area of the wire.

Equate the two expressions for power: $ I^2 R = \sigma \epsilon A (T^4 - T_s^4) $. Substitute $ R = \rho \frac{L}{A_c} $ into this equation, and note that the surface area $ A $ of the wire is $ \pi d L $, where $ d $ is the diameter of the wire.

Rearrange the equation to solve for the diameter $ d $. The cross-sectional area $ A_c $ is related to the diameter by $ A_c = \frac{\pi d^2}{4} $. Substitute this into the equation and simplify to isolate $ d $.

Substitute the known values: $ I = 18.0 \ \mathrm{A} $, $ L = 1.0 \ \mathrm{m} $, $ T = 3100 \ \mathrm{K} $, $ T_s = 20 + 273 = 293 \ \mathrm{K} $, $ \epsilon = 1.0 $, $ \rho $ (resistivity of tungsten), and $ \sigma = 5.67 \times 10^{-8} \ \mathrm{W/m^2 \cdot K^4} $. Solve for $ d $ symbolically before substituting numerical values.

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Key Concepts

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

Stefan-Boltzmann Law

The Stefan-Boltzmann Law states that the power radiated by a black body per unit area is proportional to the fourth power of its absolute temperature. This law is crucial for understanding how the tungsten wire will lose energy through radiation as it heats up. The formula is given by P = εσA(T^4 - T₀^4), where P is the power, ε is the emissivity, σ is the Stefan-Boltzmann constant, A is the surface area, T is the temperature of the object, and T₀ is the surrounding temperature.

Ohm's Law

Ohm's Law relates the voltage (V), current (I), and resistance (R) in an electrical circuit, expressed as V = IR. In the context of the tungsten wire, knowing the current flowing through it allows us to determine the resistance, which is influenced by the wire's material properties and dimensions. The resistance can be calculated using the formula R = ρ(L/A), where ρ is the resistivity, L is the length, and A is the cross-sectional area of the wire.

Heat Transfer and Thermal Equilibrium

Heat transfer refers to the movement of thermal energy from one object to another, and thermal equilibrium is reached when two objects at different temperatures exchange heat until they reach the same temperature. In this problem, the tungsten wire must reach a temperature of 3100 K while losing heat through radiation. The balance between the heat generated by the current and the heat lost through radiation will determine the required diameter of the wire to maintain this temperature.

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