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Akshay B.
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Physics (Thermodynamics)
TutorMe
Question:

We know that we can focus sunlight to reach high temperatures and use it for example to burn paper.Given a large enough lens can we theoretically do the same with starlight? What about light from planets? There are alternative ways to answer this question using both thermodynamics and optics.

Akshay B.
Answer:

First,lets use the Second Law of Thermodynamics.Imagine the lens as a Carnot Engine transferring heat from the Sun(source at temperature 5800K) to paper(sink at room temperature).Keeping the practical limitations aside the efficiency of this process can be easily calculated using Carnot's formula.What is important here is that the temperature of the sink cannot be higher than the source,so the maximum temperature you can reach by a lens is the temperature of the sun. This implies you can do the same with starlight but not with the light from planets as they are at a much lower temperature,except Venus. Another way to look at this is to observe that we are looking at extended objects making extended images. If a planet were making a point image it would have infinite temperature (remember from the Stefan Boltzmann law for a finite power area $$A\rightarrow 0$$ implies $$T\rightarrow \infty$$).

Algebra
TutorMe
Question:

Given three complex numbers $$z_1,z_2,z_3$$ on the unit circle.What is the orthocenter of the triangle formed by the three numbers?

Akshay B.
Answer:

$$z_1+z_2+z_3$$. This calculation can take a lot of time if we try to calculate the meeting points of the altitude using conventional methods. However a quick way is to see that we know that the circumcenter is at the origin and the centrois is $$\frac{z_1+z_2+z_3}{3}$$.The centroid divides the line connecting the orthocenter and the circumcenter in the ratio 2:1. This immediately gives the desired result.

Physics
TutorMe
Question:

Estimate the number of photons in the observable universe. The whole universe is isotropically filled with photons from a very early time.Because of the expansion of the universe these photons have red-shifted and are now in the microwave region.This radiation behaves like the emission from a perfect blackbody. Given the temperature of this cosmic microwave background (CMB) radiation 2.73K. Take the mean frequency of the CMB to be around 150 GHz Remember the stellar photons contribute orders of magnitude less to the radiation density compared to the microwave background.

Akshay B.
Answer:

We need to calculate the energy density of the microwave radiation.Assume photons travelling from an area A moving with speed c .We can picture a cylinder with area of cross section A and length c.The energy in this cylinder is nothing but the energy per second by the area A at temperature T=2.73 K. Now we can use Stefan Boltzmann law to calculate the power as $$\sigma A T^4$$.The energy density hence is this power divided by the volume of the cylinder Ac. using the mean frequency the number density of photons n can be calculated by using $$nh\nu=\frac{\sigma A T^4}{Ac}$$. The observable universe is a sphere of radius 39 billion light years.A quick calculation gives an estimate of the total photons in the universe.

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