Question

Question 6 There are two glowing orbs emitting black body radiation. The first has radius rand is emitting power P, while the second has radius 5r and is emitting 4 times the power. What is the ratio of the first and second orbs' temperatures? If one of the orbs is glowing red and the other is glowing yellow, which one is red?

Answer 1

for the first orb, power emitted is

$P=\sigma r^2 T_1^4$

T1 is the temperature of first orb

For the second orb, power emitted is

$4P=\sigma (5r)^2 T_2^4$

T2 is the temperature of the second orb

Divide the first equation by the second equation

$\frac{1}{4}=\frac{T_1^4}{25 T_2^4}$

Hence

$\frac{25}{4}=\frac{T_1^4}{T_2^4}$

Hence the ratio of the first and second orb's temperatures is

$\frac{T_1}{T_2}=\left(\frac{25}{4} \right )^{1/4}=(6.25)^{1/4}=1.58$

first orb is at higher temperature.

The wavelength is inversely proportional to temperature. Red has higher wavelength than yellow. Hence the second orb is red and first orb is yellow.