Light from an ideal spherical blackbody 15.0 cm in diameter is analyzed using a diffraction grat

step 1 Over here we have a light coming from a black body being passed through a diffraction grading. G

Light from an ideal spherical blackbody 15.0 cm in diameter...

Light from an ideal spherical blackbody 15.0 $\mathrm{cm}$ in diameter is analyzed using a diffraction grating having 3850 lines/cm. When you shine this light through the grating, you observe that the peak-intensity wavelength forms a first-order bright fringe at $\pm 11.6^{\circ}$ from the central bright fringe. (a) What is the temperature of the blackbody? (b) How long will it take this sphere to radiate 12.0 $\mathrm{MJ}$ of energy?

Key Concepts

Stefan–Boltzmann Law

The Stefan–Boltzmann law quantifies the total energy radiated per unit surface area of a blackbody across all wavelengths per unit time (power) as directly proportional to the fourth power of the blackbody's absolute temperature (P = ?AT^4). This principle is crucial for determining the rate of energy loss in radiative systems, and in this context, it is used to estimate the time needed for the blackbody to radiate a specified amount of energy.

Blackbody Radiation

Blackbody radiation refers to the theoretical radiation emitted by an idealized object that absorbs all incident electromagnetic radiation and re-emits energy solely as a function of its temperature. This concept underpins much of thermal radiation theory and is used to derive important laws such as Wien's displacement law and the Stefan–Boltzmann law, which describe the spectral distribution and total intensity of the emitted radiation.

Diffraction Grating Equation

The diffraction grating equation, d sin? = m?, relates the wavelength of light to the grating spacing (d), the diffraction order (m), and the angle (?) at which constructive interference occurs. This concept is essential in spectroscopy for determining wavelengths from measured diffraction angles, allowing one to infer the spectral characteristics of a light source.

Wien's Displacement Law

Wien's displacement law states that the wavelength at which a blackbody emits radiation most intensely (?max) is inversely proportional to its absolute temperature (T). The constant of proportionality, known as Wien's displacement constant, is used to calculate the temperature when the peak wavelength is known, thus connecting the thermal and spectral properties of blackbody radiation.

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