Physics

Physics

Problem

The power PP radiated by a perfect black body is thought to depend only on its surface area AA, absolute temperature θ\theta and the Stefan constant σ\sigma. Which proportionality is permitted by dimensional analysis? (Here CC is a dimensionless constant.) a) P=C σA2θ4P = C\,\sigma A^{2}\theta^{4} b) P=C σθ4AP = C\,\dfrac{\sigma \theta^{4}}{A} c) P=C σAθ4P = C\,\sigma A \theta^{4} d) P=C σAθ2P = C\,\sigma A \theta^{2}
Intuitive explanation

Dimensional analysis lets you find how physical quantities must combine based on their units alone. Power has dimensions of energy per time [ML2T−3][M L^2 T^{-3}], while area, temperature, and the Stefan constant each have their own dimensions—your job is to find what combination of AA, θ\theta, and σ\sigma produces exactly those dimensions. The Stefan constant σ\sigma is the key: it's designed precisely so that multiplying it by area and a power of temperature gives power, which is why this problem has a unique solution.

Intuitive explanation

Dimensional analysis lets you find how physical quantities must combine based on their units alone. Power has dimensions of energy per time [ML2T−3][M L^2 T^{-3}], while area, temperature, and the Stefan constant each have their own dimensions—your job is to find what combination of AA, θ\theta, and σ\sigma produces exactly those dimensions. The Stefan constant σ\sigma is the key: it's designed precisely so that multiplying it by area and a power of temperature gives power, which is why this problem has a unique solution.

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