Question

Two spheres ‘S1’ and ‘S2’ have same radii but temperatures are ‘T1’ and ‘T2’ respectively. Their emissive power is same and emissivity is in the ratio 1 : 4. Then the ratio ‘T1’ to‘T2’ is

• A. 1 : 2
• B. 2 : 1
• C. √2 : 1
• D. 1 : √2

Answer 1

Answer: (C)

√2 : 1

Answer 2

The emissive power can be represented by the Stefan-Boltzmann law:
E = σ * ε * T4
Given that the emissive power is the same for both spheres S1 and S2, and the emissivity ratio is 1:4, we can write:
E1 = E2
σ * ε1 * T14 = σ * ε2 * T24
ε1 * T14 = ε2 * T24
Since the ratio of emissivity (ε1 : ε2) is given as 1:4, we can substitute ε1 = x and ε2 = 4x, where x is a constant:
x * T14 = 4x * T24
Dividing both sides by x, we get:
T14 = 4T24

$\frac {T_1^4}{T_2^4}$ = 4
Taking the fourth root of both sides, we have:
T1 = √2 * T2
Therefore, the ratio of T1 to T2 is (C) √2 : 1.