Stefan's Law of Radiation states that the radiation energy of abody is proportional to the fourth power of the absolute temperature T of a body. The rate of change of this energy ina surrounding medium of absolute temperature M is thus (dT)/(dt)=k(M^4 - T^4) where k > 0 is a constant. Show that the general solution of Stefan's equation is ln|(M+T)/(M-T)|+2tan^(-1)(T/M)=4M^3 kt+c where c is an arbitrary constant.

Zoagliaj 2022-07-30 Answered
Stefan's Law of Radiation states that the radiation energy of abody is proportional to the fourth power of the absolute temperature T of a body. The rate of change of this energy ina surrounding medium of absolute temperature M is thus
d T d t = k ( M 4 T 4 )
where k > 0 is a constant. Show that the general solution of Stefan's equation is
ln | M + T M T | + 2 tan 1 ( T M ) = 4 M 3 k t + c
where c is an arbitrary constant.
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Answers (1)

Lillianna Mendoza
Answered 2022-07-31 Author has 16 answers
d T d t = k ( M 4 T 4 )
d T M 4 T 4 k d t
1 ( M 2 + T 2 ) ( M 2 T 2 ) d T = k d t
1 M 4 1 ( 1 + T 2 M 2 ) ( 1 T 2 M 2 ) d T = k d t
We have,
1 ( 1 + n ) ( 1 n ) = 1 2 × ( 1 + n ) ( 1 n ) ( 1 + n ) ( 1 n ) = 1 2 [ 1 1 n + 1 1 + n ]   w h e r e   n = T 2 M 2
Our equation becomes
1 2 M 4 [ 1 1 T 2 M 2 + 1 1 + T 2 M 2 ] d T = k d t
1 2 ( 1 M 2 T 2 + 1 M 2 + T 2 ) d T = k d t
Integrating both sides, we shall get
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