The emissivity of tungsten is 0.350. A tungsten sphere with radius 1.50 cm is suspended within a large evacuated enclosure whose walls are at 290.0 K. What power input is required to maintain the sphere at 3000.0 K if heat conduction along the supports is ignored?

Verified step by step guidance

1. Identify the relevant physical law: The problem involves thermal radiation and can be solved using the Stefan-Boltzmann law, which states that the power radiated by a black body is proportional to the fourth power of its absolute temperature. The formula is given by $P = e \sigma A T^4$, where $P$ is the power radiated, $e$ is the emissivity, $\sigma$ is the Stefan-Boltzmann constant ($5.67 \times 10^{-8} \, \text{W m}^{-2} \text{K}^{-4}$), $A$ is the surface area of the sphere, and $T$ is the absolute temperature of the sphere.
2. Calculate the surface area of the sphere: The surface area $A$ of a sphere is given by $A = 4\pi r^2$, where $r$ is the radius of the sphere. Substitute $r = 1.50 \, \text{cm}$ (convert this to meters before substituting).
3. Calculate the power radiated by the sphere at 3000.0 K: Substitute the values of $e$, $\sigma$, $A$, and $T = 3000.0 \, \text{K}$ into the Stefan-Boltzmann law to find the power radiated by the sphere at its operating temperature.
4. Calculate the power radiated by the sphere at 290.0 K: Since the enclosure is at 290.0 K, the sphere also radiates power at this temperature. Use the same formula, substituting $T = 290.0 \, \text{K}$, to find the power radiated by the sphere at the temperature of the enclosure.
5. Determine the net power input required: The net power input required to maintain the sphere at 3000.0 K is the difference between the power radiated by the sphere at 3000.0 K and the power radiated at 290.0 K. Subtract the power at 290.0 K from the power at 3000.0 K to find the net power input.

Key Concepts

Emissivity

Emissivity is a measure of a material's ability to emit thermal radiation compared to that of a perfect black body, which has an emissivity of 1.0. It ranges from 0 to 1, with lower values indicating less efficient radiation. In this scenario, tungsten's emissivity of 0.350 means it emits only 35% of the thermal radiation that a black body would emit at the same temperature.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law states that the power radiated by a black body is proportional to the fourth power of its absolute temperature, expressed as P = εσAT^4, where P is the power, ε is emissivity, σ is the Stefan-Boltzmann constant, A is the surface area, and T is the temperature in Kelvin. This law is crucial for calculating the power required to maintain the tungsten sphere at a higher temperature than its surroundings.

Heat Transfer

Heat transfer refers to the movement of thermal energy from one object or substance to another, occurring through conduction, convection, or radiation. In this problem, conduction is ignored, so the primary mode of heat transfer is radiation, which is significant at high temperatures. Understanding how heat transfer works is essential for determining the power input needed to keep the tungsten sphere at 3000.0 K.