Chapter 2: Problem 3

Using Wien's law and the Stefan-Boltzmann law, explain the color and intensity changes that are observed as the temperature of a hot, glowing object increases.

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Question: Explain the color and intensity changes of a hot, glowing object as its temperature increases by using Wien's law and the Stefan-Boltzmann law. Answer: As the temperature of a hot, glowing object increases, its color shifts from red to blue due to Wien's Law, which states that the wavelength of maximum radiation decreases with increasing temperature. Additionally, its intensity increases significantly due to the Stefan-Boltzmann Law, which states that the power emitted by the object is proportional to the fourth power of its temperature.

Step by step solution

Understanding Wien's Law

Wien's Law describes the relationship between the temperature of a blackbody and the wavelength at which it emits maximum radiation. This law can be written as: \(\lambda_{max} = \dfrac{b}{T}\) where \(\lambda_{max}\) is the wavelength of maximum radiation, \(T\) is the temperature of the blackbody in Kelvin, and \(b\) is the Wien's displacement constant, which is approximately \(2.898 \times 10^{-3}\) m.K.

Understanding Stefan-Boltzmann Law

The Stefan-Boltzmann Law describes the relationship between the temperature of a blackbody and the intensity of the radiant energy it emits. This law can be written as: \(P = A\sigma T^{4}\) where \(P\) is the power emitted by the blackbody, \(A\) is its surface area, \(\sigma\) is the Stefan-Boltzmann constant, which is approximately \(5.67 \times 10^{-8}\) W.m\(^{-2}\).K\(^{-4}\), and \(T\) is the temperature of the blackbody in Kelvin.

Relating Wien's Law to the change in color

As the temperature of a hot, glowing object increases, the wavelength of maximum radiation \(\lambda_{max}\) decreases according to Wien's Law. Since color is related to the wavelength of light, the color of the object changes with the change in the wavelength. Lower wavelengths correspond to the blue end of the visible spectrum while higher wavelengths correspond to the red end of the spectrum. Therefore, as the temperature increases, the color of the glowing object shifts from red to blue.

Relating Stefan-Boltzmann Law to the change in intensity

As the temperature of the hot, glowing object increases, the power emitted by the object increases as well, according to the Stefan-Boltzmann Law. The increase in power is proportional to the fourth power of the temperature, which means that even small temperature increases will result in significant increases in intensity. In summary, as the temperature of a hot, glowing object increases, its color changes from red to blue due to Wien's Law, and its intensity increases significantly due to the Stefan-Boltzmann Law.

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Using Wien's law and the Stefan-Boltzmann law, explain the color and intensity changes that are observed as the temperature of a hot, glowing object increases.

Short Answer

Step by step solution

Understanding Wien's Law

Understanding Stefan-Boltzmann Law

Relating Wien's Law to the change in color

Relating Stefan-Boltzmann Law to the change in intensity

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