Thermal Equilibrium of Lens and Disk in Solar Projection

[nicu15]

Lens & Disk

The image of the sun is projected on the disk using a lens of radius r<<d and focal length f<<d (d is the distance between the sun and the disk.. The magnification of the lens is M`f/d, so that the image completely fills the disk R`=MR. When the disk reaches thermal equilibrium, what is its temperature?

I don't really know where to start, so a hint would help a lot.

Thanks

 

[Simon Bridge]

How would you do the problem if there was no lens?

 

[nicu15]

I've already done that part.

Computed the flux in and compared it with the flux out of the disk. got a temperature of $T_{disk}=T\sqrt{\frac{R}{d}}$ where d is the distance from the sun to the disk and R is the radius of the sun.

 

[Simon Bridge]

How would you do the problem if there was no lens?

I've already done that part.

... good.

So what does the lens do in terms of the solar flux that gets to the disk?

 

[paranormal]

for your question. The thermal equilibrium of the lens and disk in solar projection is an interesting concept to explore. In order to determine the temperature of the disk in thermal equilibrium, we need to consider the energy balance between the incoming solar radiation and the heat emitted by the disk. The lens and disk system acts as a sort of "solar collector", focusing the sun's energy onto the disk and converting it into heat.

To find the temperature of the disk, we can use the Stefan-Boltzmann Law, which states that the energy emitted by a blackbody is proportional to its temperature to the fourth power. In this case, we can assume that the disk is a blackbody, as it absorbs all incoming radiation.

We also need to consider the energy absorbed by the lens, which will contribute to the temperature of the system. This can be calculated using the lens' transmittance and the amount of solar radiation it receives.

In order to reach thermal equilibrium, the energy absorbed by the disk and lens must be equal to the energy emitted by the disk. This will result in a balance of energy and a stable temperature.

Therefore, to find the temperature of the disk, we can use the following equation:

Energy absorbed by disk + Energy absorbed by lens = Energy emitted by disk

We can then solve for the temperature of the disk using the Stefan-Boltzmann Law and the known values of the disk's radius, focal length, and magnification.

I hope this helps to guide you in finding the temperature of the disk in thermal equilibrium. It is a complex system with many factors to consider, but with the proper calculations and understanding of the laws of thermodynamics, we can determine an accurate result. Best of luck in your research!