You're right, I had taken the wrong limits of integration. Sorry for the late reply by the way, I had an exam this morning.
Now continuing with the equations, let's start from the beginning. Assuming ##\chi_1=\omega^2_p/\omega##, then we can substitute in the Kramers-Kronig relation as...
Thanks for your reply, Jason. Just from the form of the equations, if I had ##\chi_1=-\omega_0/\omega##, then ##\chi_2=\pi\omega_0/2##, as when taking the limits only the evaluation on zero will lead to two non-zero identical terms. Here I have no problem evaluating the limits of the integral...
I'm kind of confused on how to evaluate the principal value as it's a topic I've never seen in complex analysis and all the literature I've read so far only deals with the formal definition, not providing an example on how to calculate it properly. Therefore, I think just understanding at least...
Well I managed to solve it and I got that both the average energy and length follow a Fermi-Dirac like distribution. I think I'll post the solution during the weekend in case anyone finds it useful.
Don't worry, with your explanation I better understood the meaning of the terms in the exponential and I think I see more clearly how to deal with these kind of systems. So then my idea about considering the tension ##\tau## for the linear case was correct since, as you mentioned, it is part of...
Indeed, the term ##pV_S## is for the pressure and volume, but since the general formula was derived for a 3D recipient I was thinking about converting it to the one-dimensional case ##pV_S\rightarrow \tau L##. However, it also makes more sense that you mention to obtain the tension as ##dZ/dl##...
Since the view is 3D, you should indeed solid angles to calculate the angle of vision. First consider the case were the astronomer is outside the well. In this case, he sees the 100% of the sky (assuming you call 100% seeing the whole half hemisphere on where they're standing). So the solid...
Homework Statement
Consider a polymer formed by connecting N disc-shaped molecules into a onedimensional chain. Each molecule can align either its long axis (of length ##l_1## and energy ##E_1##) or short axis (of length ##l_2## and energy ##E_2##). Suppose that the chain is subject to tension...
Well I checked similar procedure and I managed to advance the following:
While I don't know if it's really useful, if we apply mechanical equilibrium before adding the charge, it's straightforward to find that ##\rho_s=\rho_l##, where ##\rho_s## and ##\rho_l## are the volumetric densities of...
Homework Statement
A long straight cylinder with radius ##a## and length ##L## has an uniform magnetization ##M## along its axis.
(a) Show that when its flat extreme is placed on a flat surface with infinite permeability (i.e. a ferromagnet), it adheres with a force equal to:
$$F=8\pi a^2 L...
Homework Statement
A conductor sphere of radius R without charge is floating half-submerged in a liquid with dielectric constant ##\epsilon_{liquid}=\epsilon## and density ##\rho_l##. The upper air can be considered to have a dielectric constant ##\epsilon_{air}=1##. Now an infinitesimal...
Thanks, I'll check if we got it in the library after the morning lectures and I'll update you if I find the solution.
Edit: I checked the book, they have the proofs I needed. As for the last one, I found it here in page 4: http://mutuslab.cs.uwindsor.ca/schurko/introphyschem/handouts/mathsht.pdf
Homework Statement
Let x, y and z satisfy the state function ##f(x, y, z) = 0## and let ##w## be a function of only two of these variables. Show the following identities:
$$\left(\frac{\partial x}{\partial y}\right )_w \left(\frac{\partial y}{\partial z}\right )_w =\left(\frac{\partial...
Thank you for your time, I really appreciate it. Indeed I also checked from Jackson and Greiner and I read that I was free to choose ##F(r,r')## so that ##G(r,r')## is zero on the surface. After a couple of exchanges it turns out we were right: the Green function only depends on the geometry of...
Thank you for clearing this up. It makes sense too, since the Green function is purely geometrical.
However, when I asked the assistant of the course, he told me the Green function must also satisfy the boundary conditions of the potential in the first place, so the one I wrote is wrong (I...
You're right, I was confusing the boundary conditions of the potential with the ones of my Green function. However, would the Green function be the same even if the two semi-infinite planes are at different potentials? I understand the Green function must vanish on the surface, but I'm not sure...
Homework Statement
We have two semi-infinite coplanar planes defined by z=0, one corresponding to x<0 set at potential zero, and one corresponding to x> set to potential ##V_0##.
a) Find the Green function for the potential in this region
b) Find the potential ##\Phi(r)## for all points in...
I worked out a similar case where the cylinder is not grounded but the line charge makes the cylinder an equipotential surface with respect to it. I do this by applying the method of images again and applying the boundary conditions ##\phi(r\rightarrow \infty)=0## and ##\phi(r=R)=V_0##.
From...
Sure. The ##r_1## and ##r_2## are the distances of the line charges to the observation point. Using the law of cosinus, we can express this distance in terms of the radial distance ##\rho## in cylindrical coordinates as ##r_1^2=\rho^2+a^2-2\rho a cos\phi##. I don't have the drawing, but imagine...
Homework Statement
Find the electric potential of an infinitely long cylinder shell of radius ##R## whose walls are grounded, when in its interior a line charge, parallel to the cylinder, is placed at ##r=a## (with ##a<R##) and that has a lineal charge density ##\lambda##.
Homework Equations...
I have actually considered transitioning into the world of Machine Learning and Data Science as a second option, seeing that I have a talent for programming and I actually finished my first masters while doing applied maths. This has actually helped me a lot in my research which has a good...
I never thought about it that way, but you've got a really interesting point. You're right, the fact of finishing a PhD says a lot about the type of person, especially in undeveloped countries where less than 1% of the population manage to get one.
Also never thought about that perspective...
In general I rarely studied during undergrad since I grasped the concepts inmediatly, mastering the topic while doing the homework exercises. However, I noticed that I managed to learn fast since I always found creative ways to connect what I just learnt with what I already knew, so at the end...
Well seeing your GPA is above 3.0 (which is the minimum required for grad school), you could compensate it by getting very strong recommendation letters and highlighting your experience in CERN (which is not very common at bachelor's level), so I think you should give it a try if you're...
Hello. I'm currently beginning my 2nd semester of a dual Msc/PhD in Physics in the top university of my home country, but lately I've been struggling with self-doubt and uncertainty about the job prospects and whether research is actually for me.
To give you some background. Since I was in high...
This is one of the problems I'm currently working on but understanding how to deduce the Feynman rules for this case would give me a better idea on how to do it for more general cases besides ##\phi^4## theory (which is the example commonly covered in books like Peskin and Greiner).
1. Homework...
Thanks. That was one the articles I found but it only deals with the calculation of the angular component (p. 15), but the radial part is just given without further development, and I don't know to arrive at that expression without making the analogy to the 3D case.
Homework Statement
Find the eigenfunctions and eigenvalues of the isotropic bidimensional harmonic oscillator in polar coordinates.
Homework Equations
$$H=-\frac{\hbar}{2m}(\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial...