College Physics Questions and Answers

When light hits an atom (I will use a carbon atom for simplicity), if it is not in the absorption and/or emission spectrum of carbon, it will simply pass through without interacting with the atom. Whereas if it is in the absorption or emission spectrum it will be absorbed and either re-emitted or it will decay to become thermal energy.
However, if there are a lot of (carbon)atoms in close proximity (like in a block of coal), the light will (obviously) not pass through it no matter where it is on the visible spectrum. Why does this happen?

Suppose there exists an alpha particle in the nucleus. Within a radius of 2 femtometer, the dominating force is the nuclear force but beyond this radius, the Coulomb force becomes the effective dominating force due to it's longer range force. Outside this radius is a daughter nucleus.
So let's suppose the alpha particle moves beyond the effective range of the nuclear force,
the electric potential energy due to the force between the alpha particle and the daughter nucleus is given by as
$U_B=\frac{1}{4\mathrm{\pi <!--\; \pi \; -->}{\mathrm{\epsilon <!--\; \epsilon \; -->}}_0}(\frac{2(Z-<!--\; -\; -->2)e^2}{R}).$
Why is there the term
$2(Z-<!--\; -\; -->2)?$

Importance of Schrodinger equation
Louis de Broglie suggested that for microparticles like electrons, wave-like properties can be applied in order to explain some phenomena. Schrodinger wrote down an equation, a wave equation, describing these waves. What I do not understand is why is Schrodinger's contribution so important; if the concept of wave-like property of an electron was known already, then why writing a mere wave equation was an important step?

Why is the alpha, beta or gamma decay of an unstable nucleus unaffected by the chemical situation of an atom, such as the nature of the molecule or solid in which it is bound? The chemical situation of the atom can, however, have an effect on the half life in electron capture. Why is this so?

De Broglie Wavelength interpretation
I've just started learning about the double slit experiment (just in the short appendix section in Schroeder's Thermal Physics), and I'm extremely confused by this one thing:
In it, out of basically nowhere he pulls out the De Broglie equation, that λ = h/p.
I've studied double slit diffraction before, and I've been trying to connect them in order to understand what this wavelength actually means.
In double slit diffraction, when the wavelength is larger, the diffraction "stripes" that form on the wall appear further apart. They also appear larger.
y = ${\displaystyle \frac{m\mathrm{\lambda <!--\; \lambda \; -->}L}{d}}$ (approx, considering the distance to the screen is really large and thus almost parallel rays (drawn out waves) can have a path difference and interfere)
If we were to make the wavelength extremely small, that would mean that anything a little off-center would interfere, so the smaller the wavelength, the closer together the "stripes" on the wall would be.
Now, when we connect the 2 equations, this means that the faster the electrons are moving (the smaller their wavelength) the more places they will interfere on the wall, and therefore there will be a lesser distance between adjacent places where the electrons hit (bright spots) and places where they don't (dark spots).
The way I'm interpreting this is that the smaller the "wavelength" of the electron, the more the probability it has to have been in different places at the same time, that is, the less we can know its position. That's why more stripes will appear on the detecting screen because there are more positions which the electron could've been in, and since its technically in all of them at the same time while it travels, it can interfere with itself more.
Is this interpretation correct? Does a faster momentum (a smaller wavelength) mean that the electron literally is at more places at the same time while it travels from the electron gun, through the slits, and to the wall? Thank you!

Electric potential energy after nuclear fission
We have a uranium-236 nucleus that fissions into two equal fragments, and I'm supposed to find the electrical potential energy just as the two fragments split apart. No other information is provided.
I am very confused about this problem for two reasons. First, U-236 is electrically neutral, with exactly 92 protons and 92 electrons. After it splits into two equal fragments, why would there be a charge on the resulting pieces? Secondly, electric potential energy of two point charges is inversely proportional to the distance r between the particles. But since the question asks for the potential energy immediately after fission, r=0 and potential energy goes to infinity.
I'm appreciate any guidance!

Which has a larger frequency the 4th line of Balmer series or the 4th line of the Lyman series ?
Provide an explanation

What is the half-life of Uranium-238?

Suppose we are given a mechanical frame consisting of two points. How can we prove that assuming any initial conditions there is an inertial frame of reference in which these points will be in a static plane?

Binding Energy and Stability in Nuclear fission
I've read that particles in nuclear fission disintegrates into two particles with higher binding energy, and in this process energy is released. Now I am just trying to understand this by common sense, but I am not able to. Because the particle with lower binding energy makes particles of total net higher binding energy, meaning that it takes more energy for the particles with higher binding energy to be formed from the particle with lower binding energy. Hence, shouldn't energy actually be absorbed in this process? Could someone explain where is my reasoning wrong?

Can the astronaut tell at what speed is he moving? and how do you define speed in that case when you have no outside references. Can you refer to the speed relative to the empty space?

Radioactive element X has a half-life of 30 days. A rock sample contains 4 grams element X when it forms. How many half-lives will have elapsed in 90 days? How much of the original amount of X will be unchanged after 90 days?

Origin of the de Broglie Equation
I was curious about the famous $p=\mathrm{\hslash <!--\; \hslash \; -->}k$ equation. In high school I think you are just exposed to this equation with the explanation of "something something matter waves." But early in a undergraduate QM course you solve the time-independent Schrodinger's equation for a free particle in 1D and get the following solution:
$\frac{-<!--\; -\; -->{\mathrm{\hslash <!--\; \hslash \; -->}}^2}{2m}\frac{{\mathrm{\partial <!--\; \partial \; -->}}^2}{\mathrm{\partial <!--\; \partial \; -->}{x}^2}\mathrm{\psi <!--\; \psi \; -->}=E\mathrm{\psi <!--\; \psi \; -->}⟹<!--\; ⟹\; -->\mathrm{\psi <!--\; \psi \; -->}=e^{\pm <!--\; \pm \; -->ikx}$ and $E=\frac{{\mathrm{\hslash <!--\; \hslash \; -->}}^2{k}^2}{2m}$
where I believe $k$ is just defined by the $E(k)$ equation. And then do we just realize that the $E(k)$ is the classical $p^2/2m$ equation if we set $p=\mathrm{\hslash <!--\; \hslash \; -->}k$, and thus we have "derived" $p=\mathrm{\hslash <!--\; \hslash \; -->}k$ or do we "know" $p=\mathrm{\hslash <!--\; \hslash \; -->}k$ beforehand and this just confirms it?

In stars such as the Sun, what prevents the whole thing exploding at once? Why is the nuclear fusion happening slowly? I can only assume that something about the fusion is fighting the gravity and slowing the fusion down and when that process is done gravity starts the fusion process again.

We already have hydrogen bomb, why controllable nuclear fusion so difficult? And why controllable necessary? If we can detonate a tiny hydrogen bomb， we can collect the energy, like laser nuclear fusion do? Why didn't see any research of collider of coil gun? Beside LNF and Tokamak, using collide, we can let reaction happen in a space far from fragile device.

A reference frame on a rotating wheel is not an inertial frame of reference because it is not homogenous (depending on where we are, the body feels different inertial forces acting on them).
Could you provide me a similar example of a reference frame that is not isotropic?

What electron transitions account for the Balmer series?

Whether Compton effect depends on the frame of reference or not. The wavelength change is a thing, but does not depend on frame of reference since it depends only on θ. But if I travel at a speed comparable to velocity of light, will there be any changes?

The wavelength of emitted radiation when an electron jumps orbits in the Bohr atomic model is given by
1/$\mathrm{\lambda <!--\; \lambda \; -->}$ = $R_H$ ($\frac{1}{n_f^2}$ - $\frac{1}{n_i^2}$) $Z^2$
But that of X-Ray emission is given by
1/$\mathrm{\lambda <!--\; \lambda \; -->}$ = $1.1\times <!--\; \times \; -->{10}^7$ ($\frac{1}{n_f^2}$ - $\frac{1}{n_i^2}$) $(Z-<!--\; -\; -->1{)}^2$
Why is there a difference? Why aren't X-Rays treated like any other emissions?

Uncertainty Principle in 3 dimensions
I'm trying to understand how to write Heisenberg uncertainty principle in 3 dimensions. What I mean by that is to prove something of the form $f(\mathrm{\Delta <!--\; \Delta \; -->}{p}_x,\mathrm{\Delta <!--\; \Delta \; -->}{p}_y,\mathrm{\Delta <!--\; \Delta \; -->}{p}_z,\mathrm{\Delta <!--\; \Delta \; -->}x,\mathrm{\Delta <!--\; \Delta \; -->}y,\mathrm{\Delta <!--\; \Delta \; -->}z)\ge <!--\; \ge \; -->A$
This is what I got: The unknown volume that a single particle can be in is $\mathrm{\Delta <!--\; \Delta \; -->}V=\mathrm{\Delta <!--\; \Delta \; -->}x\mathrm{\Delta <!--\; \Delta \; -->}y\mathrm{\Delta <!--\; \Delta \; -->}z$. The uncertainty in the size of the momentum is $\mathrm{\Delta <!--\; \Delta \; -->}p=\sqrt{\mathrm{\Delta <!--\; \Delta \; -->}{p}_x^2+\mathrm{\Delta <!--\; \Delta \; -->}{p}_y^2+\mathrm{\Delta <!--\; \Delta \; -->}{p}_z^2}$
Now this is where I get stuck. In my textbook, for the 1d case, they used De-Broglie equation for connecting the uncertainty of the particle wavelength and its momentum along the $x$-axis. But does De-Broglie equation is correct per axis or for the size of the vectors?
Thanks for you help