a. The power radiated by a black-body is given by the Stefan-Boltzmann law, which states that P = σAT^4, where σ is the Stefan-Boltzmann constant, A is the surface area of the sphere, and T is the absolute temperature. For a sphere, the surface area A = 4πR^2.

For the first sphere, we have P = σ4πR^2T^4.

For the second sphere, which has a radius of 2R, the surface area is A = 4π(2R)^2 = 16πR^2.

So, the power radiated by the second sphere when its absolute temperature is T is P' = σ4π(2R)^2T^4 = 4σ4πR^2T^4 = 4P.

b. If the second sphere radiates power P, we can set up the equation P = σ4π(2R)^2T'^4, where T' is the absolute temperature of the second sphere.

Solving for T', we get T' = (P / (σ16πR^2))^(1/4) = (P / (4σ4πR^2))^(1/4) = T / 2^(1/4).

So, the absolute temperature of the second sphere when it radiates power P is T / 2^(1/4).

Question

a. The power radiated by a black-body is given by the Stefan-Boltzmann law, which states that P = σAT^4, where σ is the Stefan-Boltzmann constant, A is the surface area of the sphere, and T is the absolute temperature. For a sphere, the surface area A = 4πR^2.

For the first sphere, we have P = σ4πR^2T^4.

For the second sphere, which has a radius of 2R, the surface area is A = 4π(2R)^2 = 16πR^2.

So, the power radiated by the second sphere when its absolute temperature is T is P' = σ4π(2R)^2T^4 = 4σ4πR^2T^4 = 4P.

b. If the second sphere radiates power P, we can set up the equation P = σ4π(2R)^2T'^4, where T' is the absolute temperature of the second sphere.

Solving for T', we get T' = (P / (σ16πR^2))^(1/4) = (P / (4σ4πR^2))^(1/4) = T / 2^(1/4).

So, the absolute temperature of the second sphere when it radiates power P is T / 2^(1/4).

Knowee AI · Accepted Answer

a. The power radiated by a black-body is given by the Stefan-Boltzmann law, which states that P = σAT^4, where σ is the Stefan-Boltzmann constant, A is the surface area of the sphere, and T is the absolute temperature. For a sphere, the surface area A = 4πR^2.

For the first sphere, we have P = σ4πR^2T^4.

For the second sphere, which has a radius of 2R, the surface area is A = 4π(2R)^2 = 16πR^2.

So, the power radiated by the second sphere when its absolute temperature is T is P' = σ4π(2R)^2T^4 = 4σ4πR^2T^4 = 4P.

b. If the second sphere radiates power P, we can set up the equation P = σ4π(2R)^2T'^4, where T' is the absolute temperature of the second sphere.

Solving for T', we get T' = (P / (σ16πR^2))^(1/4) = (P / (4σ4πR^2))^(1/4) = T / 2^(1/4).

So, the absolute temperature of the second sphere when it radiates power P is T / 2^(1/4).

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