Power Radiated From a Copper Cube

In summary, "Power Radiated From a Copper Cube" discusses the phenomenon of thermal radiation emitted by a copper cube when heated. The study examines the cube's temperature, surface area, and the principles of blackbody radiation to quantify the power radiated. It highlights the relationship between temperature and radiation intensity, demonstrating how the cube's material properties influence its emissivity and overall energy output. The findings contribute to understanding heat transfer mechanisms in materials, with potential applications in engineering and materials science.
  • #1
Randomized10
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Homework Statement
What power is radiated from a 340°C copper cube 1.0cm on a side? Assume an emissivity of 1

A. 0.76 W
B. 1.8 W
C. 3.4 W
D. 8.0 W
E. 19 W
Relevant Equations
$$\frac{dQ}{dt}=e\sigma AT^4$$
##e## is emissivity
##\sigma## is the Stefan-Boltzmann constant, ##5.67*10^{-8} W m^{-2} K^{-4}##
A is the surface area
T is the temperature
##\frac{dQ}{dt}## is the rate of heat transfer or radiated power

At first glance this appeared to be an easy problem, just plug in the values and go, so that's what I did. ##e=1##, ##\sigma=5.67*10^{-8} Wm^{-2}K^{-4}##, ##A=6*(1cm)^2=6*(0.01m)^2=6*0.0001m^2=0.0006m^2=6*10^{-4}m^2##, and ##T=340^{\circ}C=613.15K##. After plugging in the values I got ##\frac{dQ}{dt}=1*5.67*10^{-8}*6*10^{-4}*613.15^4=3.402*10^{-11}*1.4134*10^{11}=4.808W##, but that isn't one of the possible answers no matter how you round it. The answer key says that the correct answer is 19 W, but I don't know how to get there. I tried working out the math using Celsius instead of Kelvin, and got ##\frac{dQ}{dt}=0.4546W##, which isn't right either. Any insights as to what I'm doing wrong would be appreciated.
 
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  • #2
My calculation agrees with yours. This is not the first time that the correct solution does not match any of the offered answers. I suggest that you show your answer to the person who assigned you this problem because he/she/ze/they needs to know.
 
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  • #3
I agree with @kuruman. Your answer of 4.8 W looks correct. The answer of 19 W corresponds to an edge length of 2.0 cm.
 
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  • #4
Ok, I'll talk to the professor. Thank you!
 

FAQ: Power Radiated From a Copper Cube

What is the formula for calculating the power radiated from a copper cube?

The power radiated from a copper cube can be calculated using the Stefan-Boltzmann law, which states that the power radiated per unit area of a black body is proportional to the fourth power of its absolute temperature. The formula is given by: P = εσAT^4, where P is the total power radiated, ε is the emissivity of the material (for copper, it is approximately 0.03), σ is the Stefan-Boltzmann constant (approximately 5.67 × 10^-8 W/m²K^4), A is the surface area of the cube, and T is the absolute temperature in Kelvin.

How does the size of the copper cube affect the power radiated?

The size of the copper cube affects the total surface area, which directly influences the total power radiated. As the size of the cube increases, the surface area increases, leading to a higher total power output, assuming the temperature remains constant. However, the power radiated per unit area remains constant, as it depends on the temperature raised to the fourth power.

Does the temperature of the copper cube influence the power radiated?

Yes, the temperature of the copper cube significantly influences the power radiated. According to the Stefan-Boltzmann law, the power radiated increases with the fourth power of the absolute temperature. This means that even small increases in temperature can lead to substantial increases in the power radiated.

What role does emissivity play in the power radiated from a copper cube?

Emissivity is a measure of an object's ability to emit thermal radiation compared to a perfect black body. For copper, the emissivity is relatively low (around 0.03), meaning it is not very efficient at radiating heat compared to materials with higher emissivity values. This low emissivity means that a copper cube will radiate less power than a black body at the same temperature.

Can the power radiated from a copper cube be measured experimentally?

Yes, the power radiated from a copper cube can be measured experimentally using devices such as thermocouples or infrared cameras to monitor temperature changes, along with a calorimeter to measure heat loss. By knowing the emissivity and surface area, the experimental data can be used to calculate the power radiated using the Stefan-Boltzmann law.

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