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1. Electrostatic Energy of a solid sphere with cavity

This is for $r>R$ so I have to also find $E(r<b)$, $E(b<r<R)$ ? Then The U will be, $$U = \frac{(R^3 - b^3)^2\rho^2 4 \ pi}{18ε_0} \int_{R}^{\inf}1/r^2dr + \frac{e_0}{2} \int_{b}^{R}\frac{\rho (r - b^3)}{3ε_0}dr$$ Where $E = \frac{\rho (r - b^3)}{3ε_0}, (b<r<R)$
2. Electrostatic Energy of a solid sphere with cavity

I tried to use $W = ε_0/2 \int E^2d\tau$ for all space. So I find that $E = \frac{(R^3 - b^3)\rho}{3ε_0r^2}$ where $\rho$ is the charge denisty. So from here when I plug the equation I get something like $$W = \frac{(R^3 - b^3)^2\rho^2 4 \ pi}{18ε_0} \int_{?}^{\inf}1/r^2dr$$ Is this...
3. I Maximum $d_p(t_e)$ value and its meaning

So the photon sended from $50c/H$ cannot reach the observer due to the expansion of the universe but later on when universe slows down the photon catches up the same proper distance at $0.1c/H_0$ ? I see.
4. I Maximum $d_p(t_e)$ value and its meaning

In cosmological models the relationship between proper distance to a galaxy at the emission and absorption times can be written as $d_p(t_e)(1 + z) = d_p(t_e)$ In this case in most cosmological models we get a maximum value for the $d_p(t_e)$. This maximum value can be also seen from...
5. I SR and Hubble Law

I understand it thanks for your reply
6. I SR and Hubble Law

This question will sound mostly stupid but anyways. We see that galaxies have some velocity due to the hubble law. Lets take an object that has a recessional velocity of $v$. In SR case assuming the universe is nearly flat, can we say that the galaxy gains mass relative to us ? I guess in...
7. I Scale factor from Friedmann's equations

At this point you can assume that $a(t) ∝ t^q$ where $q$ is just a number. So we have ¨$$\ddot{a} = q(q-1) t^{q-2}$$ $$a^{-3} = t^{-3q}$$ hence $$q (q-1)t^{q-2} = Ct^{-3q}$$ $q - 2 = -3q$ $q = 1/2$ so $a(t) ∝t^{1/2}$
8. I Scale factor from Friedmann's equations

The OP is talking about the derivatiion of the $1 + z = 1/a(t_e)$. In that integral we are taking $a(t_e)$ or $a(t_0)$ as constants because the time between two wave crests is too small, hence we can take then out. In this problem we are solving for a general time.
9. Comp Sci Trapezoidal rule for a set of data

Yes that was on my mind distance =  #the initial value for k in range(N): d = abs(1/2 * h * (function[a + k*h] + function[a + (k+1) * h])) distance.append(distance[-1] + d) It should be like this. Thansks for the tips.
10. Verlet Algorithm

I dont think energy will be conserved in the RK method. Well yes you are right, I just wanted to print them. For instance If the code runs for binary system and not for an normal orbit, we will need kinetic and potential energy values of the system. In this case avg values will not be usefull I...
11. Comp Sci Trapezoidal rule for a set of data

Thanks for your reply. I ll try to look at it again. Its possible that I have made some mistakes..
12. Dirac Delta Function

Yes I was being lazy to put those sign.
13. Dirac Delta Function

Yes I have seen it. $$\int_{-\infty}^{\infty} dx f \left(x\right) \delta \left( g \left(x \right) \right) = \Sigma f(x_i)/g'(x_i)$$ ? But I am not sure why this is the case
14. Dirac Delta Function

There arent any. We need to find I
15. Dirac Delta Function

If the question was $$\int_{∞}^{∞}dxf(x)δ((x - x_1)) = ?$$ The answer would be $f(x_1)$ So the delta function has two roots, I searched the web and some books but I am not sure what approach should I use here. I guess theres sometihng happens when $x_1 = -x_2$. So I am not sure what...
16. I Testing 2 body simulation

It passed the earth sun test. But I was more about wandering cases like binary star etc. How can assign reasonable initial conditions for such systems ? Now it came to my mind that I can try to simulate each planet im the Solar system and investigate their energy
17. I Testing 2 body simulation

I created a 2 body simulation Is there some special conditions where I can test their trajectory and check my code ?
18. Verlet Algorithm

Indeed. Also the method is not so perfect. Thanks for the help
19. Verlet Algorithm

Yes I was careful about that "-." lines are potential energy "--" lines are kinetic energy and other is total energy. My findings Potential energy: -5.30517849127762e+33 Kinetic Energy: 2.6525789562766106e+33 Total energy: -2.6525995467214404e+33 And these values from a site...
20. Verlet Algorithm

I guess I solved it ( The question claims that we should find a non circular orbit, but near elliptical one ). Here is the graph of the orbit Now I ll try to calculate the total energy of the system if it matches with the real values great!
21. Verlet Algorithm

$$F = m \frac{d^2\vec{r}} {dt^2} = -\frac{MmG\vec{r}} {r^3}$$ so $$\frac{d^2\vec{r}} {dt^2} = -\frac{MG\vec{r}} {r^3}$$
22. Verlet Algorithm

I searched online but they are using acceleration... I guess Verlet algorithm is also not much accurate... I did not understand the idea .. I was trying to solve a "one" body problem by using RK method but it seemed dead end so I was trying to do it with verlet but I couldnt understand the...
23. Comp Sci Cometary Orbits

Not really, I guess it also does not matter at this point. It seems that the RK method is not the best choice to calculate a trajectory for an object.
24. Verlet Algorithm

I am trying to understand the verlet algorithm but I am kind of stuck. I guess first we are findind the $v(t + 1/2h)$ then we are leaving it there and starting a loop for 8.78 ? Also I did not understand the meaning of the equation 8.78 ? We are never using $v(t + 3h/2)$ ? Or in the...
25. Comp Sci Cometary Orbits

So I Should change the position of the rx. So you think that my code is correct ? I changed my version 2 code.
26. Comp Sci Cometary Orbits

In second order case we should rewrite the equation in terms of 2 first order DE's. So I wrote, $$dx/dt = wx$$ $$dwx/dt = -GMx/r^3$$ and $$dy/dt = wy$$, $$dwy/dt = -GMy/r^3$$ Now I guess theres two ways to do it in 4th order RK method. I would either do it component by component or just in...
27. Comp Sci Low Pass Filter, plotting a graph

I guess I undederstand the idea, I created some other simulations and they work well.. Its hard to do it on the paper..
28. Comp Sci Low Pass Filter, plotting a graph

thats kind of sad
29. Comp Sci Low Pass Filter, plotting a graph

Sorry I mean true. Well it runs fine but there could be mistakes in those plots .. Maybe I should have get another plot..Idk
30. Comp Sci Low Pass Filter, plotting a graph

I wrote this code, but I am not sure its ture or not from numpy import arange from math import floor from pylab import plot, show Vout = 0 #inital condition t_i = 0 #initial time t_f = 10 #final time N = 100000 h = (t_f - t_i) / N RC_values = [0.01, 0.1, 1] def f(Vout, t): if floor(2 *...