( 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 ). But in real case when heated a ‘black body’ can radiate
depending upon the temperature to which it is heated. This is known as ‘black
body radiation’.
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
\[P_{1} = \sigma A T_{1}^{4}\]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 2 )
Similarly at final
temperature T2 the Stefan’s law can be written as
\[P_{1} = \sigma A T_{1}^{4}\]
. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 )
Taking ratio of eqn (
2 ) and ( 3 ) we get
\[\frac{P_{1}}{P_{2}}
= \frac{\sigma AT_{1}^{4}}{\sigma A T_{2}^{4}}\] . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 4 )
But
2 T1 = T2 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( Given )
Putting this value in
equation ( 4 ) we get ,
\[\frac{P_{1}}{P_{2}}
= \frac{\sigma AT_{1}^{4}}{\sigma A 2 T_{1}^{4}}\] . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .( 5 )
\[\frac { P _ { 1 } } { P _ { 2 } } = \frac {1} { 16 }\]
P 2 =
16 P1 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . ( Answer )
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