MHT CET · Physics · Thermal Properties of Matter

The maximum wavelength of radiation emitted by a star is 289.8 nm. Then intensity of radiation for the star is (Given : Stefan’s constant = $5.67 \times 10^{- 8} W m^{- 2} K^{- 4}$ , Wien’s constant, $b = 2898 μ m K)$

A $5.67 \times 10^{- 12} W m^{- 2}$
B $10.67 \times 10^{14} W m^{- 2}$
C $5.67 \times 10^{8} W m^{- 2}$
D $10.67 \times 10^{7} W m^{- 2}$

Verified Solution

Answer & Solution

Correct Answer

(C) $5.67 \times 10^{8} W m^{- 2}$

Step-by-step Solution

Detailed explanation

Given, maximum wavelength,

λ_{m} = 289.8 n m = 28.98 \times 10^{- 9} m

= 2898 \times 10^{- 10} m

Stefan’s constant,

σ = 5.67 \times 10^{- 8} W m^{- 2} K^{- 4}

Wien’s constant, b = 2898

μ m k

= 2898

\times 10^{- 6} m K

According to Wien’s displacement law, the maximum wavelength is given by

λ_{m} = \frac{b}{T} \Rightarrow T = \frac{b}{λ_{m}}

…. (i)
Substituting given values in Eq. (i), we get

T = \frac{2898 \times 10^{- 6}}{2898 \times 10^{- 10}} = 10^{4} K

…. (ii)
According to Stefan’s law the energy radiated from a source is given by

E = σ A e T^{4}

…. (iii)
Where, A = area of source
e = emissivity (value between 0 to 1)
The intensity of radiations emitted is equal to energy radiated from a given surface area, i.e.,

l = \frac{E}{A} = σ e T^{4}

[From Eq.(iii)]
As e is very small , so

l = σ T^{4}

Substituting the value of T from Eq.(ii) in Eq.(iv), we get

l = σ {(10)}^{4} = 5.67 \times 10^{- 8} \times 10^{16}

[∵ σ = 5.67 \times 10^{- 8} (g i v e n)]

= 5.67 \times 10^{8} W m^{- 2}

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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The maximum wavelength of radiation emitted by a star is 289.8 nm. Then intensity of radiation for the star is (Given : Stefan’s constant = 5.67×10-8Wm-2K-4 , Wien’s constant, b=2898μmK)

Step-by-step Solution

The maximum wavelength of radiation emitted by a star is 289.8 nm. Then intensity of radiation for the star is (Given : Stefan’s constant = $5.67 \times 10^{- 8} W m^{- 2} K^{- 4}$ , Wien’s constant, $b = 2898 μ m K)$