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Question

A point source of heat of power P is placed at the center of a spherical shell of mean radius R. The material of the shell has thermal conductivity k. If the temperature difference between the outer and the inner surface of the shell is not to exceed T., then the thickness of the shell should not be less than


A

2πR2kTP

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B

4πR2kTP

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C

πR2kTP

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D

πR2kT4P

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Solution

The correct option is B

4πR2kTP


Explanation for correct option

Step 1: Given data

  1. It is given that, a point source of heat of power P is placed at the center of a spherical shell of mean radius R.
  2. The material of the shell has thermal conductivity k. If the temperature difference between the outer and the inner surface of the shell is not to exceed T.

We have to find the thickness of the shell.

Step 2: Formula used.

We will find the thickness of the shell by using the term of power radiated per unit area per second in conduction.

  1. The point P is the amount of heat obtained by the source per unit of time, this heat is dispersed across the inside surface of the spherical shell, which has an area of 4πR2.
  2. So, Rate of flow of heat P=kAθx
  3. Here, k is overall heat transfer coefficient, A is heat transfer area of shell, θ is temperature difference and x is the thickness of shell.

Step 3. Calculate the thickness of shell.

By applying the given condition in the above formula, we get,

Rate of generation of heat =rate of heat transfer

P=kAθx

Here,

A=4πR2

θ=T

So,

P=4πR2Tx

x=4πR2TP

Therefore, the thickness of the shell is 4πR2TP

Hence option B is correct answer.


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