Question

Three very large plates of the 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 plate under steady state condition is

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

Answer 1

Answer: (C)

Let temperature of middle plate is given by, $T^{\prime }$
Use the condition of steady heat flow,
$\Rightarrow \sigma A{\left(2T\right)}^4+\sigma A{\left(3T\right)}^4=\sigma 2A{\left(T^{{}^{\prime }}\right)}^4$ , where $\sigma ,A$ represents Stefan's constant, area of cross-section.
$\Rightarrow {\left(2T\right)}^4+{\left(3T\right)}^4=2{\left(T^{{}^{\prime }}\right)}^4$
$\Rightarrow 16T^4+81T^4=2{\left(T^{{}^{\prime }}\right)}^4$
$\Rightarrow 97T^4=2{\left(T^{{}^{\prime }}\right)}^4$
$\Rightarrow {\left(T^{{}^{\prime }}\right)}^4=\frac{97}{2}{T}^4$
$T^{\prime }=(\frac{97}{2}T$