Question

Two spherical black bodies of radii 'r₁' and 'r₂'. Their surface temperature are 'T₁' and 'T₂'. If they radiate same power, then `"r"_2/ "r"_1` is ______.

Options

• `"T"_1/"T"_2`

• `"T"_2/"T"_1`

• `("T"_1/"T"_2)^2`

• `("T"_2/"T"_1)^2`

Answer 1

Two spherical black bodies of radii 'r₁' and 'r₂'. Their surface temperature are 'T₁' and 'T₂'. If they radiate same power, then `"r"_2/ "r"_1` is `underlinebb(("T"_1/"T"_2)^2)`.

Explanation:

Rate of energy radiation

`"Q"/"t"=sigma"AT"^4` i.e., power, P = `sigma"AT"^4` 

∴ A ∝ `1/"T"^4`

If body radiates same power, then

`"A"_2/"A"_1="T"_1^4/"T"_2^4`

⇒ `(4pi"r"_2^2)/(4pi"r"_1^2)="T"_1^4/"T"_2^4`

∴ `"r"_2/"r"_1=("T"_1/"T"_2)^2`