Question

Two bodies A and B have thermal emissivities of 0.01 and 0.81, respectively. The outer surface ares of the two bodies are the same. The two bodies emit total radiant power at the same rate. The wavelength ${\lambda }_B$ corresponding to maximum spectral radiancy in the radiation from B is shifted from the wavelength corresponding to maximum spectral radiation from A by $1.00\mu m$. If the temperature of A is 5802K, then consider the following statements:
(1) the temperature of B is 1934 K.
(2) ${\lambda }_B=1.5\mu m$
(3) the temperature of B is 11064K.
(4) the temperature of B is 2901K.

Which of the above statement(s) is / are true?

• A. (1) and (2) only.
• B. (2) and (3) only.
• C. (2) only.
• D. (1) only.

Answer 1

The correct option is A (1) and (2) only.

The rate of heat dissipated by sphere is given by equation
$\frac{dQ}{dt}=\sigma eA{T}^4$ ..(1)
. Since it is same for both the spheres A and B, we equate them and find the temp of B as 1934K.
Then, by Wien's displacement law, we get equation
${\lambda }_a{T}_a={\lambda }_b{T}_b$ ...(2)
Careful observation shows that wavelength of B will be more since temperature of A is more. Hence we get eqn
${\lambda }_b-{\lambda }_a=1\mu m$ ..(3)
Now, from equation 2 and 3, we can find wavelength corresponding to B.