Two spherical black bodies have radii ' r₁ ' and ' r₂ '. Their surface temperatures are ' T ₁ ' and ' T ₂ '. If they radiate same power, then r ₂ r ₁ is

Step-by-step solution

The power radiated by a spherical black body follows the Stefan-Boltzmann law: P = e A T^4 . Since emissivity e = 1 for a black body and A = 4 r^2 , the power becomes P = (4 r^2) T^4 . Given two spherical black bodies with radii r₁ , r₂ and temperatures T₁ , T₂ radiating equal power, equate their expressions: (4 r₁^2) T₁^4 = (4 r₂^2) T₂^4 Canceling the common factors and 4 yields: r₁^2 T₁^4 = r₂^2 T₂^4 Rearranging for the radius ratio: r₂^2 r₁^2 = T₁^4 T₂^4 Taking square roots of both sides gives the final relation