Recent content by Rlwe

Let us define matrix ##\mathbf{B}_n=[b_{ij}]_{n\times n}## as follows $$[b_{ij}]_{n\times n}:=\begin{cases} b_{ij} = \alpha\,,\quad j=i\\ b_{ij}=\beta\,,\quad j=i\pm1\\ b_{ij}=1\,,\quad \text{else}\end{cases}\,,$$ where ##\alpha\,,\beta\in\mathbb{R}## and ##n\geq2##. ##\mathbf{B}_4##, for...

Let ##f:[0;1)\to\mathbb{R}## and ##f\in C^1([0;1))## and ##\lim_{x\to1^-}f(x)=+\infty## and ##\forall_{x\in[0;1)}-\infty<f(x)<+\infty##. Define $$A:=\int_0^1f(x)\, dx\,.$$ Assuming ##A## exists and is finite, is it possible that ##\text{sgn}(A)=-1##?

Sorry, it isn't really a homework (maybe I shouldn't have posted it under HW help, sorry) but a problem I invented. I found this paper (p.10) which deals with a system of confocal ellipses and uses special system of coords to solve it. However, I couldn't find any reference that deals with...

I've been able to prove the following inequality $$\frac{2\pi\epsilon_0}{\log\left(\frac{b_1b_2}{a_1^2}\right)}\leq C \leq \frac{2\pi\epsilon_0}{\log\left(\frac{a_1a_2}{b_1^2}\right)}$$ but have no clue how to obtain exact value. Can someone check whether this inequality is correct and show how...

Thank you so much for your help.

Oh, I think that I know how the value Θ_0 was determinded. Since the disc itself cannot generate heat (i.e. if P=0 then we must have ΔT=0 ) therefore knowing the distribution of fictitious heat sources ρ(r') on the disc we must have $$\int_0^{2\pi}\int_0^R\rho(r')r'\,dr'\,d\phi=0\,.$$ After...

I've just received more hints from my professor as well as the answer to the problem and wow, your numerical solution is very accurate as the answer is $$\Delta T=\frac{P}{16\kappa R}\,.$$ He didn't provide a full solution but recommended to check that for the case of loop with disc the...

Yes, it does. According to Wolfram K(0)=π/2.

Okay, so with a help of WolframAlpha this integral can be written as $$\Theta_\text{loop}(r)=\frac{P}{2\pi^2\kappa}\frac{1}{|R_0-r|}K\left\{\frac{-4rR_0}{(r-R_0)^2}\right\}\,,$$ where K(u) is the complete elliptic integral of the first kind. I was also able to prove that for a case of disc kept...

It's rather obvious from the dimensional analysis that the solution must have the form c*P/κR, where c is some numerical prefactor so I don't think we are supposed to make such simplifications. Extremely high thermal conductivity in my opinion just means that we should assume that for any sort...

However, as I said, I don't know how to solve this Poisson's eq. with such boundary conditions. I'm almost sure it's not what we are supposed to do here since the problem doesn't ask to explicitly find Θ(r,z) but 'just' to find the difference ΔT.

I don't think these problems are the same. Here the problem is 3D with azimuthal symmetry so that the Poisson's eq. in cylindrical coordinates (r,φ,z) for temperature function Θ that I'd need to solve is $$\frac{\partial^2 \Theta}{\partial r^2}+\frac{1}{r}\frac{\partial \Theta}{\partial...

I don't know the solution. It was another homework. Using superposition of point sources to find heat flux density vector generated by the loop itself at any point would be rather hard imo. Maybe for some specific points (e.g. points on the axis) it is feasible.

There was another HW to prove that when the disc without the loop is kept at constant temperature ΔT relative to temperature at a point of the medium very far from the disc then the total amount of heat transferred per unit time through any closed surfuce surrounding the disc is $$P=8\kappa...

Honestly I'm not very familiar with Green's functions. So far we have only studied the method of separation of variables and I don't think we are supposed to use such a heavy mathematical machinery like in the article you linked. As I mentioned, my instructor said that there's an elementary...