If energy can neither be created nor be destroyed,what is the ultimate source of energy?

We usually say that the ultimate source of energy for earth is the Sun. So this means that sun creates energy but according to the law of the conservation of energy energy can neither be created nor destroyed, so how does the energy come into the sun? That is just an example. We know that energy is transformed from one form to another but is not created and destroyed. But that is confusing because there must be an ultimate source of energy. How is energy created in this source? If it is related to the atomic level, then how can atoms possess energy? Is the law of the conservation of energy untrue?

particle physics - What is the general definition of signal acceptance?

Suppose I have a beyond Standard Model theory and want to test it. I want to test if some experiments, say conduced in LHC, show signals of the theory.

In this case, what is "signal acceptance"?

superconductivity - Where does the phase difference come from in a Josephson Junction?

When you separate two superconductors by a thin insulating film, a current $I(t)=I_0 \sin{\theta(t)}$ flows between the superconductors, where $\theta$ is the phase difference between the superconducting order parameters. Where does this phase difference come from? Does it come from initial conditions?

Answer

The short answer is it comes from the current and/or voltage difference applied across the junction.

A longer answer should be of interest I believe, so here it goes.

The phase in the superconductor circuit stands for a redundant degree of freedom that you can never measure. This redundancy is important: it somehow means that a voltage is meaningless whereas a voltage-drop can be measured, and that a flux requires a closed loop to be measured as well. So it is just a way to discuss charges dynamics at the quantum level if you wish.

NB: I introduce the phase as a redundancy ; electrical engineers prefer to refer to flux node and voltage loop to discus the duality between current and voltage in circuit (along the duality between loop and node in 2D networks), see e.g. Quantum fluctuations in electrical circuits by M. Devoret (1997, free version easily found by Googling a bit) or the first years lectures by M. Devoret at Collège de France (in french). These nodes and loops are as unphysical as a redundant degree of freedom: at the end of the day, you measure voltage-drops and flux-around-loops, and that's it.

Now let's have a look at the current-voltage relation of a Josephson junction, picture taken from this website.

You see that there is a region of zero voltage and still a finite current in the diagram (plain curve). Of course this corresponds to the superconducting current violating the Ohm's law $V=RI$ (represented as the dashed line on the figure), since $R=0$ in superconductors. So what's happening there ? Well, the phase adapts itself such that the current flows. So we may prefer to write that the phase is given by the current-phase relation $$\varphi=\arcsin\dfrac{I}{I_{c}}$$ instead of the more common Josephson (first-)relation $I=I_{c}\sin\varphi$. These two rewritings are strictly equivalent as long as $I\leq I_{c}$.

Next the question: what's happening for currents larger that the critical one: $I\geq I_{c}$ ? Clearly, the phase $\varphi\in\left[0,2\pi\right]$ is no more a correct solution of the above relation. Nevertheless, this just means that we forgot to take into account the normal (non-superconducting) component of the current. We can nevertheless guess that a more complete current-phase-voltage relation should be $$I=I_{c}\Theta\left(I+I_{c}\right)\sin\varphi+\dfrac{V}{R}\Theta\left(I-I_{c}\right)$$ with $\Theta$ the step function. This is the current-phase-voltage relation you see on the picture: no voltage below the critical current since the phase redundancy allows for superconducting currents up to the critical current, and no supercurrent above the critical current when Ohm's law is verified.

Unfortunately this relation is never verified experimentally, since there are some non-linear effects for $I\geq I_{c}$ in real junctions. But the main picture remains: below a critical value (corresponding to the maximum number of Cooper pairs that can be formed across the tunnelling barrier) there is a super current flowing without resistance, and at higher current intensity you start opening normal conducting channels having some resistance. A even larger voltage you recover the linear Ohm's law.

To ensure that electromagnetism is not violated in quantum circuit, one needs also to verify $$\dfrac{\hbar}{2e}\dfrac{d\varphi}{dt}=V$$ which is nothing more than the Faraday's equation applied to the phase-difference of Cooper pairs, hence the charge $2e$ appearing (recall the engineers jargon of node-flux? here it has some interests). It is sometimes called the second-Josephson relation. So a voltage also can change the phase in a dynamic way. The previous discussion about current is valid also in the static / quasi-static situation, when the phase adapts itself instantaneously to the current change.

Of course, once again, everything is more complicated at the microscopic level, but I think I gave you the raw picture.

electrostatics - How to approximate trajectories and movement of two oppositely charged particles?

Imagine a single, stationary charged atomic ion, say a Lithium anion or cation (Li+ or Li-). Now imagine another a single free, oppositely charged particle--perhaps an electron or Hydrogen ion (H+)--passing by the first stationary atomic ion at a "classical" non-relativistic speed. For simplicity, imagine they are both in a vacuum and both in zero gravity and free from outside electrical noise or other forces.

What is a good equation to use to calculate the trajectories and/or movement of both of these particles, knowing the velocities and masses of each?

UPDATE:

While Ana V's answer below is very good, I'm really looking for an actual approximate answer of how close the two particles need to be to have measurable movement based on the forces. I'm not looking for a high level of accuracy. Just trying to get a sense of the scale of impact on the trajectory of the moving particle and the movement (if any) of the stationary particle as you vary the closest distance between them. Would they need to be very close to each other (say, less than a micron)? Or would the stationary ion still exert enough force at macro-scale distances (say, a meter?) to measurably change the moving particle's trajectory and/or push away the stationary ion? How close together do they need to be for the electrostatic forces to move one or both of them measurably? Thank you!

regarding the infinite cross section in Rutherford scattering

The differential cross section of Rutherford scattering blows as $\theta\rightarrow 0$. People express this fact as "The Coulombic potential is of long range". But I am seeing the opposite: $\theta \rightarrow 0$ is when the incident particle is not affected by the potential, and therefore a blow when $\theta \rightarrow 0$ means short range potential.

That was first. Secondly, I cannot see how the differential cross section is related to (some kind) of probabilities when infinities are involved.

solid state physics - Why the cooper pair do not obey the exclusion principle?

In superconductivity, which occurs in certain materials at very low temperature, electrons are linked together in cooper pair. And why the cooper pair do not?

Thank you in advance.

refraction - Does light change color on its way through a window?

Looking at the refractive index of glass, it's around $1.6$.

Then the speed of light $x$ through light should be given by $$ 1.6 = \frac{3.0\times10^8}{x}, $$ so $x$ is about $2\times10^8~\mathrm{m}~\mathrm{s}^{-1}$

The frequency is kept constant, so the wavelength must adapt to suit the slower speed, giving a wavelength of $2/3$ the original.

Does this mean that when passing through glass, say red light (wavelength $650~\mathrm{nm}$) changes to indigo ($445~\mathrm{nm}$), as $650 \times 2/3 = 433~\mathrm{nm}$, or is my logic flawed somewhere?

Answer

What do you take to define "red" light: a wavelength of $650~\mathrm{nm}$ or a frequency of $460~\mathrm{THz}$? On the one hand, this borders on being an ill-defined question, but I suppose it can be massaged into something answerable.

I would argue that frequency is more fundamental to describing the light. After all, it is the frequency that is constant throughout all this, as you noted. When a photon strikes a receptor in your eye, it doesn't matter whether it did so after just passing through glass or through vacuum - the biochemical response is dictated by the frequency/energy of the photon. Thus it would be more appropriate to say red light stays red, but the wavelength corresponding to red shifts in glass.