Question

Three very large plates of same area are kept parallel equispaced and close to each other. They are considered as ideal black surfaces and have very high thermal conductivity. The first and third plates are maintained at temperatures $3 T$ and $2 T$, respectively. The temperature of the middle (i.e., second) plate under steady state condition is $nT$. The value of $n ^{4}$ will be ______.

Answer 1

In steady state, the energy absorbed by middle plate will be equal to energy radiated by that plate.
$Q _{1}- Q _{2}= Q _{2}- Q _{3}$
According to Stefan's law,
$ \sigma A(3 T)^{4}-\sigma A(n T)^{4}=\sigma A(n T)^{4}-\sigma A(2 T)^{4}$
$\therefore \left(81 T^{4}\right)-(n T)^{4}=(n T)^{4}-\left(16 T^{4}\right) $
$\therefore 2(n T)^{4}=97 T^{4}$
$\therefore (n T)^{4}=\left(\frac{97}{2}\right) T^{4}$
$\therefore n^{4}=48.5$