Black Body Radiation

If the temperature of black body is increased by a factor of 2, the amount of energy and volume radiated increases by a factor of . . . . . . . . . .

( a ) 2               ( b ) 4               ( c ) 8               ( d ) 16

Definition

A black body is a theoretical object that absorbs all radiation that incident on its surface. As there is no reflection of light at room temperature the body is appears black ( that’s why it is called as black body ). Interestingly, when heated a ‘black body’ can radiate depending upon the temperature to which it is heated. This is known as ‘black body radiation’.

According to Stefan the power radiated from the blackbody can be determine by the formula

P = σ A T⁴  . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 )

Where

P = Power radiated from the black body in W ( J / s ) 

σ = Stefan's Constant 5.67 x 10 ^{- 8} W m ^{- 2} K ^{- 4}  .

A = Surface area of black body ( m ² )

 T = Temperature of body ( in Kelvin Scale [ K ] )

In other words we can say that the power radiated by the body is varies linearly with the forth power of its absolute temperature ( T ⁴ ) . Therefore the total energy increases so much for a relatively small increase in temperature. 

Stefan's Law ( P = σ A T⁴ )

   Problem:

 If the temperature of black body is increased by a factor of 2, the amount of energy and volume radiated increases by a factor of . . . . . . . . . .

Solution :

Let us consider that the P₁ be the power radiated from the black body in W ( J / s ) at initial temperature  T₁ ( K ).   ‘ A ’  be the  Surface area of black body  and  P₂ be the power radiated from the black body in W ( J / s ) at final  temperature  T₂ ( K ) .

At initial temperature  T₁ the Stefan’s law can be written as

P₁ = σ A T₁⁴  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 2 )

Similarly at final temperature  T₂ the Stefan’s law can be written as

P₂ = σ A T₂⁴  . . . . . . . .. . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 )

Taking ratio of eqⁿ ( 2 ) and ( 3 ) we get

 ( P ₁ / P ₂ ) = [ ( σ A T₁⁴ ) / ( σ A T₂⁴ ) ] . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 4 )

But

2 T₁ = T₂ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( Given )

Putting this value in equation ( 4 ) we get ,

( P ₁ / P ₂ ) = { [ σ A T₁⁴ ] / [ σ A ( 2 T₁)⁴ ] } . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .( 5 )

( P ₁ / P ₂ ) = ( 1 / 16 )

P ₂ =  16 P_{1   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .}  ( Answer )

Therefore we can say that if If the temperature of black body is increased by a factor of 2, the amount of energy and volume radiated increases by a factor of 16 (Answer : d)