# Simple thermodynamic proof that light from a source cannot be focused to heat an object above the source's temperature

I was thinking about the question in the title. I found the following thread, and some of the answers are making my head spin!

Is it possible to focus the radiation from a black body to make something hotter than that black body?

I am yet to delve into the world of optics (though I hope to soon) so I was hoping I could prove this proposition just using the laws of thermodynamics. However, nobody in the thread above posted the proof that I came up with.

This takes the form of a proof by contradiction. I construct a theoretical system in which the above process occurs, then analyse it to prove that it breaks the 2nd law of thermodynamics. The system is as follows:

1. A light source with a temperature of 6000K (sufficiently large to be considered a thermal reservoir)
2. An object whose initial temperature is 1000K

Proposed Process: The light from the source is focused onto the object. The object is heated to a final temperature of 7000K. For simplicity, assume no heat is transferred from the object to its surroundings during this process.

• Analysis: The change in entropy of the hot reservoir is calculated by the following equation: $$\Delta S = -Q / T(\mathrm{Hot})$$ (Equation taken from [1], please advise me if I have not applied this correctly)
• The change in entropy of the object being heated is calculated with: $$\Delta S = m C_p \ln (T(\mathrm{Hot}) / T(\mathrm{Cold}))$$ (Equation taken from [2], please advise me if I have not applied this correctly)

Substituting the conditions of my system into the above equation with a value of 100 kJ for $Q$ and $C_p = 1\:\mathrm J / ( \mathrm{kg K} )$ and $m = 1\:\mathrm{kg}$ we get the following result: \begin{align} \Delta S (\mathrm{Total}) &= \Delta S (\mathrm{Hot Reservoir}) + \Delta S (\mathrm{Object}), \\ \Delta S (\mathrm{Total}) &= - 100,000\:\mathrm{J} / 6000\:\mathrm{K} + \ln (7000\:\mathrm K / 1000\:\mathrm K) \times 1\:\mathrm J / ( \mathrm{kg K} ) \times 1\:\mathrm{kg} \\ \Delta S (\mathrm{Total}) & = -14.7 \:\mathrm J /\mathrm K \end{align}

The second law of thermodynamics states that the change in entropy for a real system must be greater than zero. Therefore, it is impossible to build a device that will focus the light from the source in such a way that it achieves the above.

Thus we have found a counterexample for this class of device, proving it cannot exist.

I realise that is a slightly different question to that in the other thread, as it was asking about the physical reasons why building such a device is not possible. This required the use of the aforementioned head spinning optics!