Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Why can we see the cosmic microwave background radiation? This radiation (CMBR) is said to have its origin at the surface of last scattering that exposed itself when the big bang universe had expanded for less than a million years.
In order to see radiation from a source, one has to be on its future light cone. In a un... | I get the feeling you think the CMBR is the flash from the explosion that was the Big Bang - like when Ripley blows up the Nostromo at the end of Alien.
If so, that's not really it. Rather, the universe was full of this radiation (heading in all directions) as it expanded. At our random position, we are now seeing phot... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/430581",
"timestamp": "2023-03-29T00:00:00",
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What happens to a radioactive material's atom when it disintegrates? Suppose you initial had radioactive $2^n$ atoms (where $n$ is an integer). Now after a number of halflives the number of left out atoms becomes 1. Now what will happen to it will it disintegrate and the leftover would be half an atom? Now if the reac... | Radioactive decay is a stochastic process. This means that there is random chance involved, so the exponential model used to represent radioactive does not say exactly how many atoms of the original substance will be left at a given time, rather it tells you the expected value of atoms remaining. If you begin with n=1 ... | {
"language": "en",
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Work done by friction on a body which is rolling on an inclined plane Why is the work done by friction zero during translational motion but nonzero when the body is rolling on an inclined plane?
| During pure rolling, at any instant of time, the point of contact between the roller and the ground will act as an instantaneous centre(the entire roller appears to rotate about that point at that instant).There is no sliding between the roller and the ground against friction.So the work done by friction is zero during... | {
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Once a black hole is formed, is there anything other than Hawking radiation which shortens its life? Hawking radiation is supposed to very slowly evaporate a black hole (terms and conditions apply :] ).
Apart from Hawking radiation, is there any mechanism or effect that can make a black hole cease to exist? Or once th... | Just FYI, you probably want to wait longer before you just answer your own question like that...
Anyway, black holes can merge thereby forming a new black hole - the two from the beginning no longer exist! Without appealing to some hand wavy argument about how energy is extracted from a black hole, instead black hole b... | {
"language": "en",
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"Iron Core" in Inductive Charigng Inductive charging used for wireless charging often faces the hindrance of being too short ranged for many cases. There appear to be some workarounds such as using a capacitor to resonate them at the same resonant frequency. Please excuse the naiviety of the question, but when looking ... |
So, why not just stick an iron core into the middle of the inductive
charging coils?
From the energy and field strength perspective, at high frequencies, used in wireless charging, magnetic cores are not required and, at higher end frequencies, just would not work.
From the coupling perspective, cores would not be ... | {
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Could you have sand pipes like water pipes? It's common knowledge that sand behaves like water when in small grains. So can you make a pipe that carries sand in the same way pipes carry water? If not, is there another way you could?
| Yes! If you go back to see how the old steam locomotives were built, one of the domes on the steam engine held sand that flowed down through a pipe were it ended near the tracks in front of the front drive wheel. The sand was added to provide traction between the steel wheel and the steel track. There was also a washer... | {
"language": "en",
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Angular velocity by velocities of 3 particles of the solid Velocities of 3 particles of the solid, which don't lie on a single straight line, $V_1, V_2, V_3$ are given (as vector-functions). Radius-vectors $r_1, r_2$ from third particle to first and second are given aswell. How could I find the angular velocity $w$ of... | The algebra is not especially nice, but it is just algebra. This is rigid body rotation, taking point 3 as the origin of coordinates,
so effectively
$$\mathbf{r}_1=\mathbf{R}_1-\mathbf{R}_3, \qquad
\mathbf{r}_2=\mathbf{R}_2-\mathbf{R}_3.
$$
We start as you suggested, and abbreviate
$$
\mathbf{v}_1=\mathbf{V}_1-\mathbf... | {
"language": "en",
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What if... you had a bowl of electrons? My chemistry teacher used to tell us that if you had a soup bowl with only electrons in it, the explosion could make you fly to Pluto. Was he right? Could this happen?
| The answer would depend how densely the electrons are packed. Let's say we have 1 kg of electrons, meaning we would have about $N = 10^{30}$ of them. For simplicity, let's approximate by arranging all of these electrons arranged in a spherical shell of radius $r=0.1$ meters. By symmetry, the voltage at the location of ... | {
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Instantaneous velocity So here’s a question I’ve been thinking of for a while. Suppose we say, “an object is having an instantaneous velocity along a particular direction ( say 10 m/s along the $x$-direction)” . Is it fair to conclude that it is traveling in a straight line along the $x$-axis? Well my opinion on this i... | The equation $\overrightarrow{v}=\frac{\Delta_\overrightarrow{x}}{t}$, with $\overrightarrow{v}$ being the velocity vector, $\Delta_\overrightarrow{x}$ the change in the position vector, and $t$ being the time passed only applies to a constant velocity. The more general equation for velocity would be $\overrightarrow{... | {
"language": "en",
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Lateral momentum of Gaussian beam A beam of light carries momentum. What fraction of this is lateral rather along the propagation direction if we assume something like a Gaussian beam?
Wikipedia claims in the entry on Gaussian beams that the Poynting vector is entirely along the z-axis. But this expression is based on ... | Firstly, Gaussian beam (and also Laguerre-Gauss beams) are solutions of the paraxial wave equation. Therefore, these beams are valid under the paraxial condition, which can be imposed by a requirement that the beam divergence angle is small:
$$ \theta_B = \frac{\lambda}{\pi w_0} \ll 1 , $$
where $\lambda$ is the wave l... | {
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Why does pair production produce an electron and positron with opposite spin? So I was trying to understand quantum entanglement and the example that was used to describe an entangled pair of particles was of an electron and positron after it is formed from a photon in pair production . So I was wondering why is it as... | The video is wrong on this point, at 4:47 he talks of a "photon spontaneoulsy creating an electron positron pair". The speaker is using a wrong example because there is no way a single photon can "decay" spontaneously , as mentioned in comments, due to energy and momentum conservation at the center of mass of the pai... | {
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Why does the bulk of a superconducting material expel magnetic field but not that of a perfect conductor? It is well-known that when a superconductor (SC) is cooled below the transition temperature $T_c$, the magnetic field passing through the bulk of the SC is completely expelled. In Zee's book on Quantum Field Theory... | I'll address the question in the title. I don't find the argumentation in the textbook convincing.
In a superconductor single electrons are not scattered at all. Therefore the electrons will alter their motion in the presence of a magnetic such the the flux is opposed and at sufficient depth into the bulk will be cance... | {
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Debye Temperature for Copper I am trying to calculate the Debye temperature, $\theta_D$, of copper using the following:
$$
\theta_D = \frac{\hbar v_s}{k_B} \left( \frac{6\pi^2N}{V} \right)^{1/3}
$$
I have the following values: $\rho = 8900$ kgm$^{-3}$, $v_s = 3800$ms$^{-1}$, atomic mass $ M_a=63.5$gmol$^{-1}$. Now, the... | There are few things going on here. The first is that you seem to be mixing units for density and the molar mass, using kg in one case, and g in the other. If you fix that, you will correctly get a number density on the order of $10^{28}$. However, you still won't find good agreement with the $~345K$ value you expect. ... | {
"language": "en",
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How did the Moon get into orbit? It's the accepted theory that Moon was formed when an object collided with Earth, throwing off a lump of matter which became the Moon.
But also, an object cannot be launched into orbit from a planet without extra force being applied once it's in space, because otherwise, either it leave... | You're right in that, absent post-collision interactions, material ejected during the impact would either leave on a hyperbolic orbit or be placed in an elliptical orbit that would have it falling back to the proto-Earth within one orbit.
However, the hypothesized collision is an extremely messy environment, and you do... | {
"language": "en",
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Classical Theory explanation of Compton Effect We all have studied in introduction to quantum mechanics about Compton Effect. In all the books I have read, it says that classical theory can't explain the shift in wavelength because the incident EM wave will oscillate the electron at the frequency of light, and the osci... | Compton scattering, unlike the photoelectric effect, can occur for a free electron. For a free electron, the classical theory can't explain the shift in wavelength. A theory of Compton scattering has to explain all observations, not just some of them, so it needs to explain the case where the electron is free.
Of cours... | {
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How is momentum conserved in this example? Suppose a sticky substance is thrown at wall. The initial momentum of the wall and substance system is only due to velocity of the substance but the final momentum is 0. Why is momentum not conserved?
| Remember Newton’s 3rd law. The change in momentum is $F \: \Delta t$ (also known as impulse). So, since by Newton’s 3rd law the forces are equal and opposite then the change in momentum must also be equal and opposite.
Therefore, Newton’s laws guarantee conservation of momentum, and to see where the momentum goes all ... | {
"language": "en",
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Why do we use the RMS but not the fourth root mean quad? Why do we use the power of $2$? What is the relation between this and having the same heat energy in both AC and DC?
| Average power $\left <\rm power \right > = \dfrac{\left <\rm \rm voltage^2 \right >}{R} = \left <\rm \rm current^2 \right > R$ so it is the mean of the values squared which you need to use.
In the example above you will see that the areas $A$ and $B$ above and below the $\left <\rm \rm i^2 \right > = \left <\rm \rm... | {
"language": "en",
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Kosterlitz-Thouless transition and correlation function I’m studying Kosterlitz transition on this book: https://tinymachines.weebly.com/uploads/5/1/8/8/51885267/kardar._statistical_physics_of_fields__2007_.pdf#page173 . At page 165 it says:” The gradient expansion applies to configuration that can be continuosly defo... | When doing gradient expansion, we are implicitly assuming $\theta(x)$ is a single valued function of $x$ and there is no discontinuity. Otherwise the gradient of $\theta$ diverges.
As the textbook says, any single-valued function can be continuously deformed to a uniformly ordered state. Here is how. Define a uniformly... | {
"language": "en",
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Relationship between strain energy function and strain or stress How one can get the strain or stress from the strain energy function ?
And if one cannot do it, what is the use of that function ?
| You can determine the 3D stress-strain equation for a material from its strain energy function. To get a certain component of the stress, you take the partial derivative of the strain energy function with respect to the corresponding component of strain. And, to get a certain component of the strain, you take the par... | {
"language": "en",
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(Fluid Dynamics) Euler's equation including gravity In fluid dynamics, we can write down the Euler's equation as
$\dfrac{\partial \mathbf{v}}{\partial t} + ( \mathbf{v} \cdot \mathbf{\text{grad}} ) \mathbf{v} = - \dfrac{\mathbf{\text{grad}} \; p}{\rho}$ .
If the fluid is in a gravitational field, we can add an extra te... | $\mathbf{g}$ in that equation is a generalized vector and its direction is undefined. It could be in any direction, depending what vector you choose to use. If you are using a standard cartesian reference frame, for $\mathbf{g}$ you would use (0, 0, -9.81).
All of the terms in that equation have undefined directions. S... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/434261",
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Can the horizon of a black hole move? Because of time dilation we cannot observe a black hole forming in a finite amount of time. For the same reason I suppose we also cannot observe the horizon moving: everything happening on the horizon takes an eternity to witness from the outside perspective.
Therefore, would a mov... | Since I found an answer to my own question, which I first only wrote as a comment, I'll put it here to wrap things up:
Just like photons don't age but still move, black hole horizons don't age but still move.
| {
"language": "en",
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Imprecision in experiments When dealing with breadboards, and electronic circuits in general, in my case finding the total (equivalent) resistance using an ohmmeter, what are the factors that make the experimental value not equal to the theoretical one?
Cables? The resistances themselves?
| The major factors are:
*
*Ohmmeter is not properly calibrated ("zeroed") to account for resistance of probe wires and clamp tips.
*Contact resistance between the probe tip and the circuit element due to the presence of solder resist residues, solder flux residues, or oxides/corrosion products.
*For old-style analo... | {
"language": "en",
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Why does increasing the volume in which a gas can move increase its entropy? Let's say we have a box with a non-permeable wall separating the box in half. There is gas on the other side of the wall. Now we remove the wall so that the gas can diffuse to the other half of the box.
It is said that the entropy of the gas ... | This is a classic example of making sure you know what your equations actually mean. You are thinking along the lines of
$$Q=T\Delta S$$
Where your change in entropy determines the heat exchange. This is not the right way to view the equation.
The actual meaning of the equation is "if you have reversible heat flow $Q$ ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/435119",
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Is Quantum Mechanics Compatible with Conservation of Information? What is exactly the law of conservation of information? In quantum mechanics we have truly random outcomes in experiments, but doesn't this randomness mean that new information is produced and the law of conservation of information is violated?
| As Dominic stated, local conservation laws hold in quantum mechanics, i.e. the conservation equation for probability density.
But in a simpler sense, "information is conserved" in that the total probably to measure an observable is always 1. More technically, if $\Psi(x)$ is a function that represents a quantum state i... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/435255",
"timestamp": "2023-03-29T00:00:00",
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What if we shine a laser to east and another to west, will they arrive at the same time? If we have two lasers based at the equator and we direct one of them east and the other west on to two screens each 10 km away, will light emitted synchronously at the lasers arrive at the same time as each other on thier respectiv... | It depends on your frame of reference.
If you are positioned by the laser on the Earth's surface you would observe the two laser beams to have the same speed and your would observe the two targets to be stationary and you would observe the light to take the same amount of time to reach each target.
On the other hand ... | {
"language": "en",
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Why does current conservation involve an arbitrary function? In section 6.1 of Peskin's quantum field theory introduction, right after equation 6.3, the four current density $j^{\mu}$ is said to be conserved because for any function $f \left( x \right)$ that falls off at infinity, we have
$$
\int f \left( x \right) \pa... | Current conservation is $\partial_\mu j^\mu=0$, not integrated over ANYTHING (note the zeroth component of $j$ is the current density, so this is equivalent to the usual statement of current conservation $\dot\rho=-\vec\nabla\cdot\vec j$). So, Peskin wants to show $\partial_\mu j^\mu$ is zero. He does it by showing its... | {
"language": "en",
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Conservation of energy in rotational motion Suppose a boy is standing on a platform which is free to rotate about an axis passing through its center. The Kinetic energy of the boy and the platform is K.
If the boy stretches his hands so that the moment of inertia of the sytem(boy + platform) gets doubled. Then I have t... | The boy is doing work against his own spin by extending his arms outwards. Taken to the differential limit, we can see that the force extending his arms has a tangential component that decreases angular velocity. Work is therefore done against his own rotational kinetic energy.
A similar example is the commonly-asked t... | {
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Is partial trace the inverse operation of Kronecker product? Computer science student here, who is interested in quantum information theory.
Suppose I have these pure states:
\begin{bmatrix}1&0\\0&0\end{bmatrix} and
\begin{bmatrix}0&0\\0&1\end{bmatrix}
The Kronecker product of these is:
\begin{bmatrix}0&0&0&0\\0&1&0&0... | It is in the sense that given any pair of states $\rho$ and $\sigma$, you have
\begin{align}
\operatorname{Tr}_2(\rho\otimes\sigma)&=\rho,\\
\operatorname{Tr}_1(\rho\otimes\sigma)&=\sigma.
\end{align}
This happens because $\rho\otimes\sigma$ represents a product states, in which the two parts of the system are independ... | {
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Where is mass density information of a Black Hole? Have read that all the information of a black hole is contained on the Event Horizon. Does that include mass density of interior as a function of position? The shape of an event horizon depends on the mass-energy-charge inside the black hole and possible effects by nea... | Are you asking about the viewpoint of the external observer, or somebody inside the black hole?
Because for somebody outside, yes everything about the inside of the black hole is frozen on the event horizon. For an external observer, everything is frozen on the event horizon, and so anything that entered the black hol... | {
"language": "en",
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Converting $\mathrm{ps/nm}$ to $\mathrm{ps}^2$ I have a dataset in the unit $\mathrm{ps/nm}$ for many different $\lambda$ which I want to convert to $\mathrm{ps}^2$.
I guess I can assume that I only deal with Gaussian bandwidths such that $1\ \mathrm{ps}$ corresponds to $1\ \mathrm{THz}$.
$$\Delta f = \frac{c\Delta ... | Since you are wanting to "convert" between two units with different physical meanings, there is not a standard way to do this. Multiplying a number with units of $\rm{ps/nm}$ by any number with units of $\rm{nm}$ will give you a number with units of just $\rm{ps}$. You would need to multiply by a number with units of $... | {
"language": "en",
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Work done by a gas In the expression for work done by a gas,
$$W=\int P \,\mathrm{d}V,$$
aren't we supposed to use internal pressure?
Moreover work done by gas is the work done by the force exerted by the gas, but everywhere I find people using external pressure instead of internal pressure.
| The work done by an expanding gas is the energy transferred to its surroundings. In effect, as the gas expands it is compressing its surroundings so the work done is the force exerted on the surroundings (i.e. the pressure of the surroundings times the area) times the distance moved.
The extreme case of this is a Joule... | {
"language": "en",
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Determining the acceleration of the Universe from a single star? It Occurs to me we might be able to find an entirely independent method of determining the Universe's acceleration using a single source.
If one was to watch a single high source consistently one should be able to simply simply watch for the change in it'... | You just need to measure the recession velocity of any astronomical source.
The proper distance to a given source $d$ is related to the comoving distance $\chi$ through:
$$ d(t) = a(t) \chi$$
where $a(t)$ is the scale factor for expansion of the universe. Then the recession velocity can be written as:
$$\dot{d} = \dot... | {
"language": "en",
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Are uncertainties higher than measured values realistic? Whenever I measure a positive quantity (e.g. a volume) there is some uncertainty related to the measurement. The uncertainty will usually be quite low, e.g. lower than 10%, depending on the equipment. However, I have recently seen uncertainties (due to extrapolat... | An uncertainty greater than the value for a known-positive value makes sense in the context of a non-gaussian credence.
If that first sentence was clear, you can stop reading. If not, let me back up. When we talk about "uncertainty", there isn't some sharp line involved. For any given range, we can express a probabi... | {
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Unpolarized Light: is it always linear or could it be elliptical? Unpolarized light is essentially polarized light in every direction i.e. there are so many waves radiating, that each wave oscillates in a different direction. Polarized light can either be linear or elliptical. Is light from the sun or a lamp all linear... | It can either be linear, or elliptical... or any other! There are not just 2 possibilities.
The terms "linear" or "elliptical" refer to the basis that we have chosen. If we choose a linear description, we can be lucky and find a beam which has only one-axis component. That's linear light. However, most beams will have... | {
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Continuity equation in the Lagrangian flow picture approach In deriving continuity equations using Lagrangian.
We consider the element of fluid which occupied a rectangular parallelopiped having its centre at the point $(a,b,c)$ and its edges $\delta a$ , $\delta b$ ,$\delta c $ parallel to the axes . At the time $t$ t... | *
*In the Lagrangian flow picture ${\bf a}\equiv(a,b,c)$ typically denote continuous labels of a fluid parcel distributed such that
$$d(\text{mass})~=~da~db~dc,\tag{2.1}$$
cf. e.g. Ref. 1.
*On the other hand ${\bf x}\equiv(x,y,z)$ typically denote the position coordinates of a fluid parcel. Therefore the mass densit... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Can any body be uniform in the universe? If I take any body in the shape of a rod and stretch that, after it reaches breaking stress it breaks at one point.
Even though we apply the same the stress on each and every part of the rod it broke at one point. If it's uniform it should break at all points because breaking st... | You should consider the implicances of your question, which is mostly a philosophical issue regarding our human subjectivity. You are implying an ideal uniformity of states which wouldn't allow quantum behavior.
That would mean that atoms should have all the exact same state before the breaking, that is, considering th... | {
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"timestamp": "2023-03-29T00:00:00",
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Is this possible to focus common light (not laser) to a small point in the wall and control it with mirrors? I want to make something like a laser projector for an experiment but with common light.
My question is: is this possible to focus a small point of common light to the wall and control it with mirrors? What comb... | "A common light" needs to be defined better. I assume you want to use an incoherent source such as an LED or an incandescent light. The size of the smallest spot to which a light beam can be focused depends on the size of the light source. Imagine, for example, trying to focus light from the Sun to a small spot. In e... | {
"language": "en",
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Finding the Eigenvalues and Eigenvectors of the Hamiltonian for three spin-1/2 particles coupled antiferromagnetically Problem
Given three spin-1/2 particles with the total spin operator $\vec{S}=\sum\limits_{i=1}^3 \vec{S}_i$ and its $z$ projection $S_z=\sum\limits_{i=1}^3 S_{z,i}$, and the Hamiltonian
$$H = J\sum\lim... | Just for fun, there's a second way to approach this problem. A generic basis state is of the form $| a_1 a_2 a_3 \rangle$ where $a_i$ is an up- or down-arrow. Let $P$ be the operator that exchanges the different spins: $P| a_1 a_2 a_3 \rangle = | a_2 a_3 a_1 \rangle$, so it's like a discrete momentum. It's easy to writ... | {
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Focal length vs working distance in an infinity corrected objective Consider the following microscope objective (infinity corrected):
Magnification $= 50 \times$
Glass Thickness $= 3.5$mm
NA $= 0.50$
WD $= 13.89$mm
Focal Length $= 4$mm
I'm confused about the difference between $f$ (focal length) and $WD$ (working dista... | You are correct that for a single lens the working distance would be the focal length. For compound lenses, like microscope objectives, you have to look at the entire optical system to figure out the working distance. The short answer to your question is that the focal length and working distance are not what you exp... | {
"language": "en",
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Do we need a small displacement to create a oscillatory motion on the spring? Do we need a small displacement to create a oscillatory motion on the spring with a mass attached to it? Whats the limit of the displacement that we can give initally to create a oscillatory motion? Is it has to be small or it can large?
| The displacement sets the initial amplitude of the oscillations, which will go down from that point. So, the greater the displacement, the greater the amplitude.
The only limit here is the sterchability of the spring.
| {
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Uncertain principle mistake? I believe I have a hole in my understanding of Heisenberg's Uncertainty Principle, but I'm not sure where it is.
1) Assume you have a source of monochromatic light, a laser, perhaps.
2) Assume it emits from a point isolated to a sphere of some small radius, e.g. 3cm.
3) Assume I can isola... | If the duration of the light is finite--that is, the laser beam is not infinitely long--then there is a non-zero uncertainty in the energy of the light. This is true regardless of the construction of the light source. In fact, the shorter the pulse, the broader the energy spectrum. This fits with the uncertainty princi... | {
"language": "en",
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Orbital parity of simple bound states in atomic and particle physics The parity operator commutes with the Hydrogen atom Hamiltonian. The energy eigenfunctions are parity eigenstates with orbital parity $(-1)^\ell$ which follows from the fact that $Y_{\ell m}(\theta,\phi)$ is an eigenstate of parity with parity eigenva... | In the central field approximation, the Hamiltonian of a many-electron atom is assumed to comprise of a large spherically symmetric part and a small spherically asymmetric component. Since the asymmetric part is small, it can be treated as a perturbation. The total angular momentum operator $\textbf{L}=\sum\limits_{i}\... | {
"language": "en",
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What is the relation between physical theory and physical law? Gravitational law was explained by Newtons theory of gravity. So a law was described by a theory. What is the theory for Newton's laws of motion?
| The laws are the theory. Newton did not explain why the gravitational force is $GMm/r^2$. He theorized this formula, and showed that it explained the solar system and apples falling from trees.
Similarly his three laws of motion, plus some assumptions about absolute space and time, are the theory of Newtonian mechanics... | {
"language": "en",
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Are there any general results about the nodes of energy eigenfunctions in higher dimensions? A well-known result of quantum mechanics is that for a single particle in one dimension in a bounding potential $V(x)$ that goes to $+\infty$ as $x \to \pm \infty$, the energy eigenfunctions are discrete and the $n$th eigenfunc... | I found the answer on Chemistry SE. Apparently, for systems of one particle in higher than one dimension, the only known general result is that the number of nodes in the $n$th eigenfuntion is $\leq n-1$, and only in 1D is the inequality always saturated. (See the answer for more info and precise term definitions.) Mor... | {
"language": "en",
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Why does gauge invariance in electrodynamics mean that there are redundant degrees of freedom? It is possible to choose different gauges in electrodynamics. I am familiar with two of them: Coulomb gauge and Lorenz gauge. Let us stick to the Coulomb gauge. It sets $$\nabla\cdot\vec{A}=0.$$ The wisdom is that with this c... | The scalar and vector potentials are gauge covariant which is to say $\Phi' \ne \Phi,\,\mathbf{A}' \ne \mathbf{A}$ where
$$\Phi'(\mathbf{x},t) = \Phi(\mathbf{x},t) - \frac{\partial}{\partial t}\chi(\mathbf{x},t)$$
$$\mathbf{A}'(\mathbf{x},t) = \mathbf{A}(\mathbf{x},t) + \nabla\chi(\mathbf{x},t)$$
The electric and magne... | {
"language": "en",
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Where does the principle of equal a priori probabilities come from in statistical mechanics? I have studied that the principle of equal a priori probabilities yields maximum entropy principle and minimum free energy principle and we can define and calculate other thermodynamic variables. However, where does the princip... | Principles ( and laws, and postulates) in physics are the equivalent of axioms in a mathematical theory. The mathematical format used to study physics is very broad . A subset of the possible solutions allowed by the mathematical axioms is picked up by the use of laws, as the conservation laws, and principles, as for ... | {
"language": "en",
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Relativistic space rocks. Are they possible? I was thinking that the Universe is full of extreme events. Colliding galaxies, exploding stars, colliding planets (like in one of models of our Moon formation), colliding black holes...
Can it be possible, that such event would accelerate some rocks to let's say $0.1c$ ?
D... | Suppose the total energy release of a typical supernova is on the order of 1044 J, for a baseline. The question then becomes, if 100% of that energy goes into hurtling a physical object, how massive would the object be if the ejection speed was 0.1c?
At 100% efficiency (which is entirely unrealistic), the mass of the ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/439315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why don't all gasses have infinite entropy? Entropy of an ideal gas is defined as the logarithm of the number of possible states the gas can have multiplied by Boltzmann's constant:
$${\displaystyle S=k_{\mathrm {B} }\log \Omega .}$$
In deriving the Maxwell-Boltzmann distribution, we initially start by counting a finit... | The distribution is continuous is exactly true only in the thermodynamic limit, in which case $E$, $V$, $N$, and $S$ are all infinite, while $E/V$, $N/V$, and $S/V$ are all finite (or, equivalently, $E/N$, $V/N$, and $S/N$ are finite). In a real-life situation in which the volume isn't infinite, the distribution is te... | {
"language": "en",
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Is the amplitude of an EM wave the combination of the electric AND magnetic fields added together? For instance, to get the TOTAL energy of an EM wave(s) or intensity you square the amplitude. But do you first add or combine the strengths of the e and m fields?
| Suppose you have two sources of Electric field, $E_1$, and $E_2$
Then $\vec{E_{tot}}=\vec{E_1}+\vec{E_2}.$
So the total intensity is
$$E^2_{tot}=(\vec{E_1}+\vec{E_2})^2=E^2_1+E_2^2+2\vec{E_1}\cdot\vec{E_2}$$
So the intensity of the sum of two electric fields is not the sum of their intensities individually.
The same t... | {
"language": "en",
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Is there a traditionally accepted threshold probability at which highly unlikely becomes impossible? Many events which by any practical definition are impossible have extremely low but nonzero probability of occurrence. For instance, the positions of oxygen molecules in a room are basically random and independent, but ... | This idea that events with probability under some small value are to be treated as impossible is known as Borel's or Cournot's law, but is not actually an accepted law, as far as I know.
It is however understandable that in applications of probability calculus, the events with extremely low probability (such as Sun exp... | {
"language": "en",
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Formula for potential energy? Conservation of energy? How would we know what formula to use for potential energy?
In my class, $mgh$ was used, but when dealing with a spring, it's ${1\over2}kx^2$. Is that because that's the elastic potential energy formula?
Also, for elastic and inelastic collisions, momentum is conse... | For conservative forces the Potential energy can be defined as $$U=-\int_{x_i}^{x_f} \vec{F} d\vec{x}$$ For example think about $$\vec{F}=mg(-\vec{i})$$
I put a minus sign because it means that I choosed upward direction as positive. If we put the equation we have
$$U = -\int_{x_i}^{x_f} mg(-\vec{i})dx\vec{i} = mg(x_f-... | {
"language": "en",
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Geodesics of anti-de Sitter space It is said that (p. 9), given the anti-de Sitter space $\text{AdS}_2$, let's say in the static coordinates
$$ds^2 = -(1 + x^2) dt^2 + \frac{1}{(1+x^2)} dx^2$$
Every timelike geodesic will cross the same point after a time interval of $\pi$. That is, if $(x_0, t_0) \in \gamma$, then $(x... | First, the statement
will cross the same point after a time interval of $\pi$
is wrong. In the cited paper the actual statement
… each timelike geodesic which intersects the $t$ axis at the point $t=t_0$
intersects that axis again at $t=t_0+\pi$.
So the $\pi$ interval refers to passing through the $x=0$, the a... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/440470",
"timestamp": "2023-03-29T00:00:00",
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Wave group and phase velocities When talking about phase velocity and group velocity they tend to be both expressed in terms of another $v$. From my understanding this $v$ comes from the wave equation, but what does it represent? And regarding phase velocity my understanding was that it is the velocity of a specific fr... | {
"language": "en",
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How does an object in space travelling at constant velocity have a net force of zero acting upon it? If the definition of balanced forces is "two opposing forces that are equal" and an object with a net force of zero acting upon it means that the forces are balanced on the object, then it should follow that an object t... | I think I understand your point, and it does make sense.
The way I see it is the following : the object you're throwing in space is firstly acted upon a force (let's call it F1). You're suggesting that since the object is traveling with a constant velocity, then the net force on that object must be 0, which is correct.... | {
"language": "en",
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Cylinder vs cylinder of double the radius roll down an incline plane, which one wins? A solid cylinder and another solid cylinder with the same mass but double the radius start at the same height on an incline plane with height h and roll without slipping. Consider the cylinders as disks with moment of inertias I=(1/2)... | Conservation of energy tells us potential energy becomes kinetic energy as the disks fall. If they roll without slipping, some energy goes into translational kinetic energy and some goes to rotational kinetic energy.
The rolling without slipping condition requires that the velocity of the disk be equal to the rotatoin... | {
"language": "en",
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Spatial part of Robertson-Walker metric The spatial part of the FRW metric can be written as $$d\Sigma^2=d\rho^2+f^2(\rho)(d\theta^2+{sin}^2\theta d\phi^2)$$ where $f(\rho)$ satisfies $$\frac{df}{d\rho}=\frac{f(2\rho)}{2f(\rho)}.$$ I am trying to derive the form of $f(\rho)$ by using a power series expansion $f(\rho)=\... | Should be:
$f(\rho) = \sum a_n \rho^n$
$df/d\rho = \sum n a_n \rho^{n-1}$
$f(2 \rho) = \sum a_n (2 \rho)^n = \sum 2^n a_n \rho^n$
You plug what above in the equation, you multiply L.H.S. and R.H.S. times the expression for $f(\rho)$ and move all the terms to the L.H.S. Then order the terms as per powers of $\rho$, e... | {
"language": "en",
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Is the normal force equal to weight if we take the rotation of Earth into account? In my physics class we were doing problems such that we set $N$ (normal force) $= mg$. I understand that by Newton's Third Law, if I exert a force on the ground, then the ground will exert an equal and opposite force on me. However, the ... | I'd like to finalize the conclusion from @AaronStevens's great answer. In the truer expression for normal force (on flat ground) that he arrives at,
$$N=mg-\frac{mv^2}{r}=m\left(g-\frac{v^2}{r}\right)\quad ,$$
Earth's rotation adds the term $\frac{v^2}{r}$ so it deviates from the expected $N=mg$. How much is the influe... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/441245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Lagrangian of EM field: Why the $B$-field term has a minus sign in front of it in the Lagrangian? I know that $L = T - U$ and that, in the non-relativistic case
$$L= \frac{1}2mv^2 - q\phi(r,t) + q\vec{v}\cdot\vec{A}(r,t).\tag{1} $$
My lecturer used the following form of the Lagrangian density to derive Maxwell's equat... | In the gauge $\phi=0$, the $E$ term is $\frac12\dot A^2$, which is kinetic energy, and the $B$ term is $(\nabla\times A)^2$, which is potential energy and therefore gets a minus sign.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/441516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Do centrifugal force and gravity differ in their effects on objects? If the type of object matters, consider the human body. If the situation matters, consider standing on the inside wall of an O'Neill cylinder compared to standing on the surface of Earth.
"Differ in their effects on objects" means: Would the object be... | General Relativity is compliant with the Strong Equivalence Principle. According to this principle:
The outcome of any local experiment (gravitational or not) in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime.
This implies that locally gravity is indistingui... | {
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"url": "https://physics.stackexchange.com/questions/441606",
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Why does the warm air rises up? Warm air has more energy than cold air. This means that according to the Einstein equation $E = mc^2$ the warmer air has a greater mass than the cold one. Why is the warm air rising, if it has a greater mass, which means that the attraction of gravity between the Earth and the warm air i... | Even though $E=mc^2$ is only for objects that are not moving and we really should be using $E^2=p^2c^2+m^2c^4$, you are conceptually correct in that objects with more energy have more mass. The issue is that c is a huge number so it takes a ton of energy to actually give something mass in a significant way. c is 299,79... | {
"language": "en",
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Relationship between freefall velocity time dilation and gravitational time dilation in a Schwarzschild metric If you drop an object into a gravitational field, is its final velocity equal to what it would have to be in flat space in order to generate the same time dilation that you get at a given radius for an object ... | Yes, this is correct. From Wikipedia:
"Time dilation in a gravitational field is equal to time dilation in far space, due to a speed that is needed to escape that gravitational field. Here is the proof.
*
*Time dilation inside a gravitational field $g$ is $t_0 = t_f \sqrt{1 - \frac{2GM}{rc^2}}$
*Escape velocity fro... | {
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Would a supersonic object without a combustion power source leave behind a contrail? Contrails, as far as I understand them, are caused by either a pressure change that forces the condensation of H2O(g) OR by the release of warm H2O from a combustion engine. Most plane contrails, I would assume, operate largely by this... | The space shuttle is an example of the phenomenon of a supersonic craft leaving behind a contrail.
The Fact that the space shuttle flies as a glider on re-entry is stated by NASA here.
The velocity of the space shuttle at different altitudes during descent is depicted here in the section "Repeat explanation of a Shuttl... | {
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If the molecular collisions are elastic will there be any dissipation in a fluid? Viscosity arises due to collisions of the molecules of one layer of a fluid with another in contact. But viscosity is a dissipative element leading to heating and dissipation. Where does it heat come from? Does it come from the molecular ... | The viscosity arises because of the elastic collisions, not in spite of them.
When temperature in the fluid rises, it is because the molecules are moving faster, due to elastic collisions. The energy is called "heat" because the motion is disorganized (random).
Its entropy has increased.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/442363",
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What is the shape of a gravitational wave form? What is the shape of a gravitational wave as it hits the Earth, particularly the time portion.
Does time start at normal speed, then slow slightly, and then return to normal speed?
Or does it start at a normal speed, slow down slightly, then speed up slightly, and then ... | A simple monochromatic gravitational plane wave has two possible transverse polarizations. If the wave is traveling in the $+z$ direction, then the metric for one of the polarizations can be written in the simple form
$$ds^2 = -dt^2 + (1+h_+)\,dx^2 + (1-h_+)\,dy^2 + dz^2$$
where the small metric perturbation $h_+$ is w... | {
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How should I imagine a multi-particle state in a free QFT? It is reasonable to think of single-particle Focks states as of plane waves. Indeed, since $|p\rangle=a^\dagger_p|0\rangle$ and $\langle x|p\rangle\sim \operatorname{e}^{ipx}$, we conclude that the state $|p\rangle$ can be thought of as a plane wave in the posi... | Take a compound system of many strongly interacting particles at $T=0$ and give it a shake. The system in an excited state can be approximately described as a set of free quasi-particles corresponding to the normal modes of the system.
| {
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Minimum separation from the spacetime interval I've been working through invariant spacetime interval questions recently, and I came across a question in my lecture notes where;
$$\Delta s^2=\Delta x^2 -(c\Delta t)^2 > 0 $$
Now it is clear to me that there is no frame where $\Delta x' = 0$ which I have already proven a... |
But since $t'$ can be equal to 0, I'm not sure where to go from
here.
$\let\D=\Delta$
You meant $\D t'$, didn't you? You were on the right track, but went
lost. Your first equation says
$$\D s^2 = \D x^2 - (c\,\D t)^2 > 0\tag1$$
Then you wrote
I am assuming that the spacetime interval is the same in every frame
w... | {
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Is there a simple way to calculate Clebsch-Gordan coefficients? I was reading angular momenta coupling when I came across these CG coefficients, there is a table in Griffith's but doesn't help much.
| It might depend on your definition of "simple".
For easy cases (low numbers, direct steps), yes, it is simple. However, it gets complicated too fast. What we do is: calculate only the easy ones, and let the rest for computers. I really encourage you to do this.
The trick is: at the top of the ladder, there is only one... | {
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Particle in a 2d box, using de broglie wavelength to find momentum of the particle Consider a proton in a 2-dimensional infinite potential box with widths, L and K.
a) Using de Broglie wavelength, find the momentum of the particle.
I find some online sources that finds the energy (not momentum) using the wavefunction ... | Momentum is a vector. The norm is given by $p=\sqrt{p_x^2+p_y^2}$.
| {
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Could quantum fluctuations spawn real matter? Would it be plausible for fluctuations in the QED vacuum to spawn actual matter (such as quarks, electrons the constituents of a hydrgen atom) given enough time and space?
| Note: People seem to think that an empty universe is an eigenstate of the full QED Hamiltonian. This is not true. The Hamiltonian is
$$H= \sum_{e^+e^-} \omega_p (b^\dagger_p b_p +c^\dagger_p c_p)+ \sum_{\rm photons}\!k\,a^\dagger_k a_k\,+\, \sum_{k+p+p'=0}(a^\dagger_k + a_k)(b_p c_{p'} + c^\dagger_p b^\dagger_{p'})$$
I... | {
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Does it make sense to view all solutions to Einstein's field equations as $g_{\mu \nu}=\eta_{\mu \nu}+h_{\mu \nu}$, for a given $h_{\mu \nu}$? I have a question about General Relativity that so far I have never been able to solve.
Let's take Einstein's field equations:
$$ R_{\mu \nu}-\frac{1}{2}g_{\mu \nu}R=\frac{8 \pi... |
Is it possible that there exist certain solutions that cannot be expressed as a Minkowski background $\eta_{\mu \nu}$ plus fluctuations $h_{\mu \nu}$?
Yes. There are links between curvature and topology, and some topologies are not consistent with a Minkowski metric. For example, closed FLRW spacetimes have the spati... | {
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When to use sine or cosine when computing simple harmonic motion For simple harmonic motion (SHM), I am aware you can start of using either sine or cosine, but I am a bit confused as to when you would start off with sine rather than cosine. I know that a sine graph starts at $y=0$ and a cosine graph starts at $y=1$. So... | "a particle starting from rest at time t=o" This is the key to why you start with cosine. So you know that instantaneous velocity is the derivative at that point. Look at a sine graph at t=0 there is a positive gradient thus the particle doesn't start at rest with sine. But with cosine at t = 0 the gradient is zero thu... | {
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Does a rock use up energy to maintain its shape? A rock sitting on land, the ocean floor, or floating in space maintains its shape somehow. Gravity isn't keeping it together because it is too small, so I'm assuming it is chemical or nuclear bonds keeping it together as a solid. If not it would simply crumble apart. So,... | The amount of work done is equal to the distance moved times the force in the direction of motion. As the rock is staying the same shape it does not need to exert energy.
You may be thinking that the rock needs to expend energy in order to hold up its heavy mass in the same way our muscles do if we hold up a heavy weig... | {
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Two People Pushing Off of Eachother, Newton's Third Law, and Unbalanced Force Different versions of this question have come up all over the internet. Usually it deals with tension in a rope or two people pushing on each other with the same force. I am trying to understand 2 people pushing each other with different forc... | The problem here stems from understanding closed systems.
The center of mass of a closed system (no external forces) does not move. This comes from the fact that Newton's third law states that any internal force applied will be canceled by another opposite and equal internal force.
The forces you and your friend are ... | {
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Commutive property of the Bra-ket notation I'm struggling when it comes to understanding the commutive properties of the Bra-ket notation in quantum mechanics. I understand how to work with constants, bra and kets. However, the second I start introducing eigen-equations such as
$$\hat{x}|x'\rangle = x'|x'\rangle$$
to s... | When writing $\hat x\vert x'\rangle=x'\vert x'\rangle$, the $x'$ is actually a number (aka a scalar) and so can be moved about like a regular number, so that
$$
\langle \psi_p\vert\hat x\vert x'\rangle =
\langle \psi_p \vert x'\vert x'\rangle = x'\langle\psi_p\vert x'\rangle
$$
because $x'\in \mathbb{R}$, much in the... | {
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Laplace transform on thermodynamics equation I'm trying to create a simple model of a single-flash geothermal plant which consist of 3 main parts (flash-separator, turbine, and condenser). Is there any way to create a transfer function using Laplace Transform (s-domain) of each parts based on the thermodynamics equatio... | The first thing you need to do is write down the time-dependent mass- and energy balance equations for the turbine (using the open system control volume version of the 1st law of thermodynamics). Then you express each the variables as the sum of a steady state part and a transient part. Then you solve for the steady ... | {
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Lorentz' Derivation of Lorentz force I have one simple question: Can someone point me to the paper where H.A. Lorentz published Lorentz force? I was digging through the usual literature in electromagnetics (Jackson, Griffiths, Thide, Sommerfeld, Stratton) for the reference on the original Lorentz paper with no success.... | What is nowadays known as the Lorentz force law was originally due to Maxwell, equation 77 in Part 2 of his 1861 paper On Physical Lines of Force (p. 482 of vol. 1 of his Scientific Papers), which in more modern vector notation looks like:
$$\mathbf{E}=\mu\mathbf{v}\times\mathbf{H}-\frac{\partial\mathbf{A}}{\partial t}... | {
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What is the function of this complicated tensioning system? I saw this arrangement for tensioning overhead cables from my train window (schematic below). Why not just have one pulley wheel leading directly to the weights? What function do the additional pulleys serve? For that matter, what are the cables for? They're c... | This is a so-called block and tackle arrangement which is often used for tensioning of overhead lines. Tensioning is required to keep a desired line geometry and, in case of contact wires, to avoid standing mechanical waves (waves in a tensioned line travel faster).
Apart from mechanical advantage, such systems often p... | {
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Why are green screens green? How/why do green screens work? What's so special about the color green that lets us seamlessly replace the background with another image and keep the human intact?
Are there other colors that work similarly?
| It's partly about how human colour vision works, partly about avoiding colours you want to keep, such as those of the actors.
Colour cameras record concentrations of red, green and blue light to mimic human colour vision. Before digital techniques, blue screens were preferred because, of the three primary colours, that... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/445158",
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Movement of fluid in a container filled with that same fluid If a cylinder with the bottom end closed and the top end open was filled with water and then dropped in a pool of water. Would the water inside the cylinder stay in the cylinder?
| You could do this experiment at home. Take a glass of water, add some food colouring und then drop it into your bathtub (fill it first with enough water, otherwise your 'container' will burst...). My prediction:
The water in the glass gets accelerated, first, together with the glass. The glass will then hit the water a... | {
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Examples of central forces on the path of orbit? In solving a problem from Goldstein (3.13), I solved for multiple properties of a circular orbit with the attractive central force where the path of orbit crosses the point of the force (at origin).
The solutions were simple enough to find, but what's been in the back o... | Are you asking purely about an orbit system? Or a general system? One I can think of is simple harmonic motion. Take a mass on a spring. The force will always act towards the origin, and the mass will go through the origin. Although the definition of a central force is one that always acts towards a fixed point (in thi... | {
"language": "en",
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Neutral buoyancy for cylinder/bladder in water A 25 liter flexible plastic bladder consists of three parts: an air chamber located on top of the bladder, a large middle water chamber, and a lower weighted area that serves as a counterbalance and ballast found below the bladder. The bladder is cylindrical in shape and i... | Since your just looking for an estimated ratio to get you close from the jump, consider the volume of air in the bag, and the weight of volume of water it is displacing. Counterweight for the weight of the water displaced and fine tune it from there.
| {
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In what context is enthalpy a convenient concept? Internal energy $U$ is clearly an important concept; the first law of thermodynamics states that for an isolated system internal energy is constant $(\Delta U=0)$ and that for a closed system the change in internal energy is the heat absorbed by the system $Q$ and work ... | Enthalpy is a particularly convenient concept when analyzing components, such as turbines, compressors, pumps, condensers, evaporators, and expansion valve, in open steady flow processes. Here are some examples for the case of an ideal Rankine power cycle where the $h$ is enthalpy:
Reversible, adiabatic turbines: $\dot... | {
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Can the second law of thermodynamics be derived from Quantum randomness? The second law of thermodynamics says that the entropy of an isolated system continuously increases. Can we say that this is due to Quantum mechanics, which continuously increases the information contained in the system by producing random numbers... | The laws of thermodynamics can be derived (or at least motivated) from statistical physics.
To do so, we assume that there is an "underlying theory" describing the microscopic physics of the system.
However, we do not know have perfect information about the state of the system (we say that we do not know the "microstat... | {
"language": "en",
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What is the length of null geodesic? There are many questions about this but none of them adresses my concrete question. If it is indeed true that for light we have $ds^2 = 0$ does that mean that in 4d spacetime total "distance" is zero for light? By distance I mean lenght of a geodesic that light moves on? Describes? ... | Spacetime manifolds are observer's manifolds, and there is no spacetime manifold all observers agree upon. However, according to the second postulate of special relativity, lightlike phenomena are observed by all observers as moving at c, so there is always observed a traveled distance > zero.
In contrast, the spacetim... | {
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Can we ever "measure" a quantum field at a given point? In quantum field theory, all particles are "excitations" of their corresponding fields. Is it possible to somehow "measure" the "value" of such quantum fields at any point in the space (like what is possible for an electrical field), or the only thing we can obser... | One must be careful not to confuse formalism for physics. What do we mean when we talk about a field? In quantum field theory (which is a formalism) a field is often represented by an operator. What does that operator represent in the physical world? The field operator is in fact used to define the observables (correla... | {
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Work-Energy Theorem for Non-Constant Mass "The net work done on an object is equal to its change in kinetic energy."
Let's say that a rocket is moving upwards while expelling gas and is thus losing mass. (Non-constant mass)
As the object loses mass, it gains even more kinetic energy.
Is the work-energy theorem still ... | I believe the problem lies with how you are stating the work-energy theorem.
Every good reference to it I can find states something along the lines of:
"The work done on a particle is equal to it's change in kinetic energy"
(emphasis mine) The word particle is extremely important here (sometimes they also use "rigid... | {
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Having trouble making sense of Einstein's thought experiment So I was reading about Einstein's thought experiment where he tries to show that simultaneous events in one frame may not be simultaneous in another frame.
So, in the given pic, light from B' reaches Mavis before light from A' and I get that because she is m... | The speed of light is constant for all observers, so if Stanley sees lightning reach A’ and B’ simultaneously, then Mavis must see it reach B’ first, as you said. The diagram shows that this occurs, from Stanley’s perspective, because Mavis is moving to the right and the light from B’ reaches her before the light at A’... | {
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Does a particle with infinite energy escape an infinite well? Currently, my modern physics class is going over particles in finite and infinite wells, general quantum formalism, and tunneling.
What happens to a particle as it gains an infinite amount of energy? Does it stay inside of the infinite well? Does it escape? ... |
Are there any issues with this question? Is it valid? Is there anything I need to define or presume before I ask it? Do I need to define the rates at which the potential of the walls go to infinity, or the rate at which the particle's energy goes to infinity?
Your question is valid, but only if you take an infinite ... | {
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Could someone explain what is a potential? In many part of physics, me talk about potential (electrical potential, gravitation potential, elastic potential...). All those definition looks very different, and I would like to know how all those quantity are related. The mathematical definition of a potential is : $F$ is ... | A scalar potential whose gradient is the force exists in many branches of physics, for example heat conduction, static magnetism, static electricity, etc. In all these cases the gradient (or negative gradient) of the potential field is the force that can "move" things from here to there.
For example, in static magneti... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Stack/Chimney Effect: a physical explanation on how height of chimney affects $\Delta P$ A fellow engineer told me that there are greenhouses which exploit the stack effect, in order to cover some or all of their electrical energy needs. This is achieved by installing small electrical generators with fans mounted on th... | The chimney only has to be high enough to contain an extended column of warm/light air. Too short and the warm air might spill out into the ceiling of the greenhouse instead of being siphoned up the chimney. Every home fireplace relies on this effect to keep smoke from filling the room.
Pressure decreases with height... | {
"language": "en",
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What is the "lowest energy"? In many textbooks I come across the term lowest energy. For example in atomic structures, electrons are placed in orbitals in order for the atom to have the lowest energy. But what is this energy? Potential- or kinetic energy or the sum of the two?
| The sum of the two.
An electronic state like an orbital is an exact or approximate solution of some time independent Schrödinger equation, i.e. an eigenstate of the hamiltonian made by a kinetic and a potential energy term. The corresponding eigenvalue is the expectation value of such hamiltonian, evaluated on the sta... | {
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Can a battleship float in a tiny amount of water? Given a battleship, suppose we construct a tub with exactly the same shape as the hull of the battleship, but 3 cm larger. We fill the tub with just enough water to equal the volume of space between the hull and the tub. Now, we very carefully lower the battleship into ... | For the interested, here is a little illustration I made once, to defend someone called 'Marilyn vos Savant'.
I didn't believe what people wrote in there against her statement, about a battleship floating in a bucket of water.
I am very happy to see, that everybody in here knows what is right.
Here is the illustration:... | {
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"source": "stackexchange",
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How is $\int \frac{d^{3}\mathbf{p}}{(2\pi)^3}\frac{1}{2\sqrt{|\mathbf{p}|^2+m^2}}$ manifestly Lorentz-Invariant? When writing integrals that look like
$$
\int \frac{d^{3}\mathbf{p}}{(2\pi)^3}\frac{1}{2\sqrt{|\mathbf{p}|^2+m^2}} \ = \int \frac{d^4p}{(2\pi)^4}\ 2\pi\ \delta(p^2+m^2)\Theta(p^0)
$$
it is often said that th... | The measure is invariant under proper orthochronous Lorentz transformations (those continuously connected to the identity, or equivalently those that do not represent a spatial reflection or time reversal) because $\Theta(\Lambda^0_\nu q^\nu) = \Theta(q^0)$ for such transformations as they preserve the sign of the temp... | {
"language": "en",
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"source": "stackexchange",
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2D global conformal transformations and the $z= \frac1w$ argument
For instance in Blumenhagen's CFT, there is a standard argument which determines that globally defined conformal transformations on the Riemann sphere where
$$l_n = -z^{n+1} \partial_z$$
is an element of the Witt algebra. In this argument we note tha... | The minus sign is not essential, so let's remove it for simplicity. An atlas of the Riemann sphere, say $M$ ( which is a 1-dimensional complex manifold), is given by two coordinate charts
*
*$z:M\to\mathbb{C}$, which covers the whole $M$ except the "infinity" $i\in M$.
*$w:M\to\mathbb{C}$, which covers the whole... | {
"language": "en",
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How can two electrons repel if it's impossible for free electrons to absorb or emit energy? There is no acceptable/viable mechanism for a free electron to absorb or emit energy, without violating energy or momentum conservation. So its wavefunction cannot collapse into becoming a particle, right? How do 2 free electron... | Your first statement is false: energy can indeed be added at will to electrons by accelerating them with electrostatic charge distributions, as for example in the case of rapidly varying radio frequency (electromagnetic) fields. Neither energy nor momentum conservation is violated in this case. Search on SLAC for more ... | {
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Rule of addition of velocities in water Is the speed of light still the same for all inertial observers in water?
If not, what are the rules of addition of velocities in water according to special relativity?
| No, the speed of light in water depends on the observer's speed. If you're at rest relative to the water, you'll see a wave with 4-velocity:
$$ u_{\mu} = \gamma_n(c, 0,0, c/n) = \gamma_n c (1, 0,0,\frac 1 n ) $$
where:
$$ \gamma_n \equiv \frac 1 {\sqrt{1-\frac 1 {n^2}}}$$
Now you can boost it any which way, and you'll ... | {
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Violating Newtons First Law! Suppose you are inside a very large empty box in deep space , floating ( i.e not touching the box from anywhere initially).The box is at complete rest.
Now you push the box forward from inside.
Now you would go backwards but the box will move forward to conserve momentum.
However since you... | Your confusion stems from the misunderstanding of the term "internal force".
Internal force in this context is not a force which acts from the geometrical inside of some body. It is internal in the sense that it results from the interaction of the parts of the system, whose collective motion (the acceleration of its ce... | {
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How can we define energy other than the definition that it's a capability to do work? It is actually a property of energy that it can do some work not an actual mean to define it because we cannot define a thing on the basis of what it is doing or what it can do.
| From my perspective, it is just a number associated with a system or object. For instance, a 2 kg ball moving with velocity 5 m/s has kinetic energy of 25 J. This number is always conserved for a system of objects (for instance 2 balls colliding, or two charges acting on each other) provided that no external forces act... | {
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} |
Is it the current which create magnetic field, or vice versa, or both? Talking about stationary magnetic field, it is said that if a conductor rotates inside the field, a current is induced. Also, I read that current (moving charges) generate magnetic field, too. How are these connected, and what's the best approach to... | Very briefly, here is how to get started:
You are right- if a conductor experiences a changing magnetic field in its vicinity (as, for example, when it is moved past a magnet in such a way that it cuts across the magnet's "lines of force") then a current will be induced to flow in that conductor. This is called FARADAY... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/449916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
What was wrong with the old definition of temperature scale in kelvin? Wikipedia's article on the recent change to the definition of the SI base units states, as the reason for changing the definition of the kelvin:
A report published in 2007 by the Consultative Committee for Thermometry (CCT) to the CIPM noted that t... | It is difficult to determine Boltzmann's constant from the temperature of the triple point. That is easier at extremely low temperatures (with statistical mechanics) or at very high temperatures (from radiation).
At ambient temperatures, the connection to Boltzmann's constant is by using gas thermometers. Very cumberso... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/449975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
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