Question

Three very large plates of same area kept parallel 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 temperature $2T$ and $3T$ respectively. The temperature of the middle (i.e. sound) plate under steady state condition is
(correct answer + 1, wrong answer - 0.25)

• A. ${\left(\frac{65}{2}\right)}^{1/4}T$
• B. ${\left(\frac{97}{4}\right)}^{1/4}T$
• C. ${\left(\frac{97}{2}\right)}^{1/4}T$
• D. ${\left(97\right)}^{1/4}T$

Answer 1

The correct option is C ${\left(\frac{97}{2}\right)}^{1/4}T$

Let temperature of middle plate is $T^{\prime }$
In steady state, energy absorbed by middle plate is equal to energy released by middle plate.
$\sigma A\left[\right(3T{)}^4-\left(T^{\prime }{)}^4\right]=\sigma A\left[\right(T^{\prime }{)}^4-\left(2T{)}^4\right]$
$2(T^{\prime }{)}^4=97T^4$
$T^{\prime }={\left(\frac{97}{2}\right)}^{1/4}T$