Question

20. If the temperature of a Black body increases by 50% then amount of radiation emitted by itin
given time interval will​

Answer 1

Answer:

Explanation:

Here,

initial  thermodynamic temperature of black body, T₁ = 100

The temperature of a Black body increases by 50%. so temperature after this increase, T₂ :

T₂  = T₁ + 50 = 100 + 50 = 150.

by taking the ratio of final temperature and initial temperature, we get

T₂/ T₁ = 150/100

T₂ / T₁ = 3 / 2

T₂ = 3/2 T₁

=> According to Stefan–Boltzmann law:

U₁ = σeAT₁⁴ ...(1)

U₂ = σeAT₂⁴ ...(2)

=> By taking the ratio of eq(1) and (2), we get

U₁/U₂ = σeAT₁⁴/ σeAT₂⁴

U₁/U₂ = T₁⁴/ T₂⁴

U₁/U₂ = T₁⁴/ (3/2 T₁) ⁴

U₁ / U₂ = 16 T₁⁴/ 81 T₁⁴

U₁ / U₂ = 16/81

U₂ = 81/16 U₁

=> Increase in percentage:

increase in % = (U₂ - U₁) / U₁ * 100%

                      =  81/16U₁ - U₁ / U₁ * 100

                      = (81/16 - 1) U₁ / U₁  * 10

                      = 81 - 16 / 16 * 100

                      = 65 / 16 * 100

                      = 4.063 * 100

                      = 406.3 %

Thus, amount of radiation emitted by black body in  given time interval will​ be 406.3 %.

                                     

Answer 2

Given that,

The temperature of a Black body increases by 50%.

Let the initial temperature is T.

Final temperature $T'=T+\dfrac{50}{100}T$

$T'=T+\dfrac{T}{2}$

$T'=\dfrac{3}{2}T$

The initial amount of energy of radiation emitted

Using formula of energy

$E_{1}=\sigma eAT_{1}^4$

The final amount of energy of radiation emitted

Using formula of energy

$E_{2}=\sigma eAT'^4$

Put the value of T'

$E_{2}=\sigma eA\times\dfrac{3}{2}T$

$E_{2}=E_{1}\times(\dfarc{3}{2})^4$

$E_{2}=E_{1}\times\dfrac{81}{16}$

We need to calculate the percentage increases

Using formula for percentage increases

$\text{percentage increases}=\dfrac{E_{2}-E_{1}}{E_{1}}\times100$

Put the value into the formula

$\text{percentage increases}=\dfrac{\dfrac{81}{16}E_{1}-E_{1}}{E_{1}}\times100$

$\text{percentage increases}=(\dfrac{81}{16}-1)\times100$

$\text{percentage increases}=406.25\%$

Hence, The percentage increases is 406.25%.