Question

Three very large plates of same area are 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 temperatures $2T$ and $3T$ respectively. The temperature of the middle (i.e., second) plate under steady state condition is :

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

Answer 1

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

Let the area of each plate be $A$.

Total power absorbed by plate $2$ is $P_{abs}=P_1+P_3$

$\therefore$ $P_{abs}=\sigma Ae(2T{)}^4+\sigma Ae(3T{)}^4$

Total power radiated by plate $2$ is $P_{rad}=P_2+P_2$

$\therefore$ $P_{rad}=2P_2=2\sigma Ae(T^{\prime }{)}^4$

At steady state, $P_{rad}=P_{abs}$

$\therefore$ $2\sigma Ae(T^{\prime }{)}^4$ $=\sigma Ae(2T{)}^4+\sigma Ae(3T{)}^4$

Or $2(T^{\prime }{)}^4=(2^4+3^4)T^4$

$⟹$ $T^{\prime }=(\frac{97}{2}{)}^{\frac{1}{4}}T$