Answer

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**Hint:**Here a body is cooled down and then heated up again. We have given the change in temperature for which it was cooled down and heated again and we have to find the total heat required from a heater by per kg of the body. By using Newton’s law of cooling we can solve the given question.

**Formula used:**

\[Q=ms({{T}_{f}}-{{T}_{i}})\]

**Complete answer:**

Here a body whose temperature was initially \[40{}^\circ C\] was cooled down till \[38{}^\circ C\] in 10 minutes. Then the body was heated again to raise its temperature from \[38{}^\circ C\] to \[40{}^\circ C\] in the same time duration that is 10 minutes.

Now according to Newton’s law of cooling, the heat which was loss during cooling the body is given as the product of its mass, specific heat of the body and the change in temperature and its formula is

\[Q=ms({{T}_{f}}-{{T}_{i}})\]

We also know that heat loss by a system is equal to the heat gain. Similarly, the heat required to raise the temperature of the body will be equal to the heat loss when it is cooled down in the same time duration. Hence the heat required to raise the temperature of the body here will be equal to the heat loss when it was cooled down. And heat loss can be calculated by Newton's law of cooling. But as we have to calculate the heat required by the heater by per kg of the body we can rewrite the above equation as

\[\dfrac{Q}{m}=s({{T}_{f}}-{{T}_{i}})\]

Where \[\dfrac{dQ}{m}\] is the heat loss by per kg of the body, s is the specific heat which is given \[2J/{}^\circ C\], \[{{T}_{i}}\] is the initial temperature which is \[40{}^\circ C\] and \[{{T}_{f}}\] is the final temperature which is \[38{}^\circ C\]. Substituting all the values we get

\[\begin{align}

& \dfrac{Q}{m}=2(38-40) \\

& \Rightarrow \dfrac{Q}{m}=2\times -2 \\

& \Rightarrow \dfrac{Q}{m}=-4J \\

\end{align}\]

Where the negative sign shows the heat loss.

Hence the heat gain or heat required from a heater by per kg of body will be 4J.

**So, the correct answer is “Option D”.**

**Note:**

Here the heat loss or gain due to surrounding is not mentioned and so we have not considered it. In case, we consider the effect due to the surrounding we have to add our answer with the heat required to keep the temperature of the surrounding constant if it was changing. Because there will be heat loss or heat flow in the surrounding also.

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