mprovod

What would be the location of circular and straight line fringes for both point and extended sources in a Michelson interferometer? How would I use the Michelson interferometer in the analysis of spectral lines? I think there is no refraction in the Michelson interferometer so how could individual spectra be separated?

A grating with N slits of spacing d is iluminated normally with monochromatic light of wavelength λ, and the p-th order principal maximum is observed at an angle Өp. The intensity at angle Ө is given by: I(α) = sin²(Nα/2)/sin²(α/2) where α=(2Π/λ)d*sin(Ө) Show from this equation that adjacent minima fall at angles Өp+αӨp and α-Өp, where αӨp = λ/(Ndcos(Өp)) Could someone please help me? I tried to differentiate the Intensity equation w.r.t. α to find what the values of the minima and maxima are, but then realised, that the separation of minima from maxima must depend on the angle Ө. I don't know how to approach this further.

Mode note: moved to HW help ——— Suppose that by some artificial means it is possible to put more electrons in the higher energy state than in the lower energy state of a two level system (this sort of situation occurs in a laser, for example). Now it is clear that this cannot be an equilibrium situation, but nevertheless, for the time that the system is in this strange state we could, if we wished, still express the ratio of the populations in the upper and lower states by some parameter we can think of as an effective temperature. (i) Show that for such a population inversion to exist, the effective temperature must be negative. (ii) Imagine I have electrons that populate two states in the normal manner at room temperature. I then somehow swap the populations (i.e. all the ones that were in the lower state go into the upper state, and vice versa). What is the new effective temperature? (iii) What is the effective temperature if I put all the electrons in the upper energy state? I'm quite puzzled by this. Could someone please explain this or tell me the formulae I need to use? Thanks a lot. I have also attached the problem set I'm trying to solve. I couldn't get questions 2.4 and 2.2 as well so I'd welcome answers or explanations of those as well. Thanks. sm-questions.pdf

Show that the thermodynamic properties of an assembly of N three-dimensional harmonic oscillators are the same as those of an assembly of 3N one -dimensional oscillators. I can easily imagine this, but how do you actually prove it?

I know that entropy (S) S = kln(Z) + kTd(ln(Z))/dT Here, Z is the partition function and k is Boltzmann constant.

An array of N 1D simple harmonic oscillators is set up with an average energy per oscillator of (m + 1)hf Show that the entropy per oscillator is given by S/N = k[(1+m)ln(1+m) - (m)ln(m)] Comment on the value of the entropy when m = 0

I've figured it out. Thanks a lot.

2.1 Consider an array of N localised spin-1/2 paramagnetic atoms. In the presence of a magnetic field, B, the two degenerate spin states split by ±μB, where μB is the Bohr magneton. (i) Derive the single particle partition function for the system. (ii) Show that the heat capacity C can be written as C = dU/dT = NkB((D/T)^2)exp(D/T)/(exp(D/T)+1)^2 (here, Kb is Boltzmann constant) and find the value of the constant A. Show that C has a peak at a temperature Tpeak = AμBB/kB where A is a numerical constant. Determine A. I think I found the answer to part (i), which I think is 2cosh(beta*μB*B), where beta is just the greek symbol. However, I can't figure out the second part, especially how to get the relation for C. Once I have that it should be fine. Thanks for help.

Why is enthalpy conserved in steady flow processes?

I'm supposed to show how to rewrite Dieterici's equation of state: P(V − b) = RTe^−a/RTV in reduced units: p(2v − 1) = t*exp(2*(1 − 1/tv)), where p=P/Pc, v=V/Vc and t=T/Tc and (Pc, Tc, Vc) is the critical point. I should further show that the equation of the inversion curve is p = (8 - t)exp((5/2) - (4/t)). I'd be greateful if someone could show me especially the first part. I'm not sure how to approach this. Thanks.

Two thermally insulated cylinders, A and B, of equal volume, both equipped with pistons,are connected by a valve. Initially A has its piston fully withdrawn and contains a perfect monatomic gas at temperature Ti, while B has its piston fully inserted, and the valve is closed. The thermal capacity of the cylinders is to be ignored. The valve is fully opened and the gas slowly drawn into B by pulling out the piston B; piston A remains stationary. Show that the final temperature of the gas is Tf = Ti/(2^(2/3)) I've been thinking about this one for hours, could someone have a look and let me know if you're successfull?