Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Why light shows its wave-like properties only when it interacts with objects with dimensions close to the wavelength of light? In Young's Double Slit Experiment, we were taught that light behaves as a wave here because the width of the slits are very close to the wavelength of light itself. But why does light behave li... | Consider an optical lens, say a magnifying glass. The light interacts with it, its path bending as it passes through. Yet the lens is far larger than the wavelength, while the individual atoms are much smaller.
It is simply that interference effects are easiest to see when the dimensions are similar, as evidenced in th... | {
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Purely resistive AC circuits Is it possible to have a purely resistive RC circuit?
Recently came across a question: A 130Ω resistor and a 40μF capacitor are connected in series to an AC source of frequency ω. For the combination to be purely resistive what must be the value of frequency ω?
Is this question right or a... | The only way the circuit can look "purely resistive" is if there is an inductor in series with the capacitor and resistor and where the frequency is such that the inductive reactance equals the capacitive reactance, or
$$\omega L=\frac{1}{\omega C}$$
$$\omega=\frac{1}{\sqrt {LC}}$$
where $\omega$ is now the resonant fr... | {
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How do Standing Waves satisfy the criteria in order to be considered as Waves? Why are standing waves even categorised as waves if they don't transfer energy from one point to another?
Waves are generally defined as disturbances that transfer energy but standing waves don't fulfil that criterion. Even though they are f... | Energy is being transported. A standing wave can be written as a superposition of a left-moving wave, which transports energy to the left, and a right-moving wave, which transports energy to the right. For a standing wave, it happens that there is no net transfer of energy to the right or to the left. But there are sti... | {
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If we free a metal plate of free electrons will it become weaker to external mechanic forces? If we free a metal plate of free electrons by means of a strong electric field will it become then weaker in case of applied external forces? To make it simpler, do free electrons have any role in connections between lattice a... | When one puts a capacitor at the ends of a battery, the metal plates on the negative side will have a lot more charge and the other plate will display a positive charge, which means a lack of electrons. So in this sense you do free electrons from one side and add them on the other.
do free electrons have any role in ... | {
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Low-energy excited states of $^{13}C$ While doing some studying for an exam in introductory nuclear physics, I stumbled upon a question I can't answer. I'm supposed to explain the ground state and the three lowest energy excited states of $^{13}C$ using the shell model. The ground state has spin and parity $1/2^-$. The... | Unfortunately there is no easy rule to unambiguously determine whether the valence neutron is in 2s1/2 or 1s1/2 orbital. The situation is even more complex if you consider the interacting shell model, which include 2-body interaction between those single-particle states. Usually if you consider some specific model spac... | {
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Is Landau theory for phase transitions valid only for "order to disorder" phase transition? In the Landau theory we assume order parameter that is equal to zero at $T>T_c$ and none zero at $T<T_c$ which is valid only for order to disorder phase transition according to my understanding.
So that is mean that I can't use... | The Landau paradigm is getting a bit outdated in my opinion (not only) . There are plenty of different phase transitions which cannot really be categorised as first or second order. Enough to look at lattice models of quantum gravity or the standard model, but in analytic models also the study of Kibir-Żurek mechanism ... | {
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Where is energy in energy density? I was learning about energy density and it seemed to be defined as the potential energy per unit volume in an electric field
$\frac{dU}{dV} = \frac{1}{2}\epsilon E^2$
But how can just the electric field have a potential energy on its own without presence of any charge? What is causing... | There are charges present. They are the cause for the electric field. Defining something based on the electric field is indirectly defining it based on the charges and charge configuration.
| {
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Why is the reflection coefficient 1 for step potentials where energy is less than the potential? Consider a potential $V(x)$ which is zero when $x<0$ and $V_0>0$ when $x>0$. Suppose there is an incident particle with momentum $p=\hbar k$ and energy $E = \hbar^2 k^2 / 2m < V_0$ coming from $x= - \infty$. Now then after ... | There are two ways of thinking about it, one is strictly mathematical (shut up and calculate, in a way), and another is maybe not very precise but rather intuitive. The first one you did correctly, $R=1$. In a potential $V=V_0$ for all $x>0$ (continuing infinitely "long" to the right) there is no way for the particle w... | {
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Does mass have an effect on Centripetal Acceleration? I am using an online simulation for a lab concerning Centripetal Acceleration. When I change the mass the graph indicates that the magnitude of the acceleration is constant. According to the Centripetal Acceleration formula: $a=v^2/r$, this is true because no mass i... | Here's a more math-y restatement of the same general answer :)
Just from your question, we see that $v^2/r = a = f/m$.
If the simulation keeps $v^2/r$ constant ($a_1$ = $a_2$), then when $m$ changes, $f$ should change. Depending how sophisticated the simulation is, it may or may not report the value of $f$.
If you solv... | {
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Occlusion angle subtended by a black hole Suppose we have a Schwarzschild black hole of mass $M$ at a distance $R$ to an observer. Due to the curvature of space, the apparent angle (or half-angle) of occlusion $\theta_a$ of the silhouette of the blackhole will be larger than the angle described by its Schwarzschild rad... | Any light ray that crosses the photon sphere will hit the horizon, and any light ray that doesn't, doesn't. So, your guess that it would be the angle subtended by the photonsphere is pretty good. However, in reality the angle is slightly bigger, since two light rays that start parallel at infinity will have bend furthe... | {
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Lepton Universality Does universality of lepton also mean that two lepton families have the same renormalized coupling if they interact with a particular particle?
| Wikipedia: “The coupling of leptons to all types of gauge boson are flavour-independent: The interaction between leptons and a gauge boson measures the same for each lepton.” [Note: Recent experiments are raising questions about this.]
Lepton universality does not apply to their Higgs couplings; if it did, they would h... | {
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Change in the mass of a star due to nuclear fussion of Hydrogen Let's suppose a star is initially composed only of hydrogen. After a certain time $t$, it has consumed the $5\%$ of its initial hydrogen, fusing it to produce helium, according to the following reaction:
$$4 H^1 \rightarrow He^4 +\Delta E$$
I'm trying to f... | If the atomic mass of $^4\text{He}$ were exactly $4$ times the atomic mass of $^1\text H$ then $\Delta E$ would be zero and the mass of the star would be constant (and we would not be here to think about this problem). However, the atomic mass of $^4\text{He}$ is slightly less than $4$ times the atomic mass of $^1\text... | {
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EM wave power dependency on frequency Does a (classical) radio wave with a given amplitude carry more power if that wave is at a higher frequency than at a lower frequency?
| Perhaps you need to ask a similar question on Amateur Radio StackExchange. There they could explain to you the observable details of how a given antenna rod and a given power from a wave generator change the output of the antenna. I will give you an idea of how the achievable output power of a radio wave changes at con... | {
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Are the physical structures in our sun of comparable complexity to those in the human brain? The writings of Rupert Sheldrake tend to provoke strong emotions, be they ridicule, curiosity, outrage, sympathy, disgust, or otherwise. While Physics SE is not an appropriate forum in which either to debunk or to promote his g... | We cannot compare the complexity of objects; we can only compare the complexity of our models for these objects.
And when we talk about models, the complexity of a model increases when our understanding of the object increases. Given enough resources and motivation, we can research an object forever.
Observations and r... | {
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A clarification on acceleration and velocity This is one of those questions which require an answer that does not take practical limitations into account. It is a theoretical physics question, perhaps. If there are any loopholes used, please explicitly state them.
If the position is known as $x(t)$ from t=0 to t=1 seco... | This question is more interesting if you refer to numerical calculations. For analytical tratement, it is simple: velocity is the derivative of position.
If you say you know $x(t)$ for $t\in[0,1]$, then velocity is $v(t)=dx/dt$. But yes, the endpoints can be more problematic. However, real numbers are so dense that you... | {
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What does an $x$ Watt bulb actually means? Let's say I have a 11W bulb in my home, connected to a 220V power supply. What exactly does it mean that this bulb is 11W? As far as I know, the Wattage is determined by the formula $W = V * I$, so it really depends on the voltage and the current. The current is determined by ... | Yes, the current depends on the internal resistance. And that resistance has been adjusted to give a current that corresponds to 11 W.
(By the way, note that we typically don't symbolise the wattage, the power, with $W$ but rather with $P$.)
| {
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Are Lie algebras of groups unique? Take for example $GL(2,\mathbb R)$, the group of $2\times2$ invertible matrices with real entries. By considering small variations from the identity, it is clear that one needs four parameters to parametrize this group, and hence we will need four "infinitesimal" generators. If I am t... | Like most mathematical objects, when we ask whether two Lie algebras are "the same" we should really be asking whether these is an isomorphism of Lie algebras between them. Since a Lie algebra is a vector space with extra structure (the Lie bracket), such an isomorphism is an isomorphism of vector spaces $f : L_1 \to L... | {
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Solution to differential equation If I have a differential equations of the form $$\frac {d^2y}{dt^2}=\alpha^2y$$
Assuming the roots of the characteristic equation is complex the solution to the differential equation is: $$y=C_1e^{j\alpha t}+ C_2e^{-j\alpha t}$$ and after that we take only the real part of the solution... | This is a common question in classical physics.
The idea is that we are really treating a problem where everything is real number-valued, but complex numbers offer convenient mathematical short-cuts.
In the present example you have a differential equation
$$
\frac{d^2y}{dt^2} = \alpha y
$$
and let's agree at the outset... | {
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Why aren't all actions conformally invariant? I am very confused about coordinate invariance of actions in classical field theories on arbitrary background spacetime or even with dynamical metric. From this question, we see that if the integrated term, namely the Lagrangian density, of the action is well-defined, i.e. ... | Note the words "conformal transformation" can mean slightly different things in different places. But conformal transformations, in the sense that it's used in conformal field theories, are not just coordinate transformations. Instead they are a simultaneous coordinate and field transformation such that the metric is l... | {
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Can the mass-shell equation be derived from the path integral formulation I've recently been trying to wrap my head around the notion of virtual particles, which as far as I understand live in quantum histories which can never be observed directly and which are not bound by certain laws of physics, but which may evolve... | Since Feynman considers point particles let us discuss this case. Yes, OP's speculations can indeed be realized. The Hamiltonian action $$\begin{align}S_H[x,p,e]~=~&\int d\tau ~L_H, \cr L_H~=~&p_{\mu}\dot{x}^{\mu}-\frac{e}{2}(p_{\mu}p^{\mu}+m^2), \end{align} $$
for a relativistic point particle has the mass-shell const... | {
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Amperian loop for different situations Why are amperian loops always taken as circles? What if we take it as a triangle or rectangle? Give an example by taking amperian loop as rectangle for current carrying straight wire and derive equation for magnetic field? How will the equation change?
| An Amperian loop can be any closed path. Ampere's law is true for any closed path.
$$\oint \vec{B}\cdot d\vec{l} = \mu_0 I_{\rm enc} $$
Whether the law is useful or not depends on whether you are able to easily calclulate the LHS (the closed line integral of the B-field around the loop), and the RHS (the total current ... | {
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How to generate electric current without a permanent magnet? The question is pretty simple:
Can we build a device that coverts mechanical work in electric current1 without employing a permanent magnet and without access to any external source of current?
The restrictions in place seem to rule out the possibility of c... | To answer this question one needs to think of a form of energy and then a device that will convert that energy into electrical energy.
Mechanical - Wimshurst machine, Van de Graaff generator, piezoelectric crystal, perhaps a self-exciting generator?
Light - Solar cell
Thermal - Thermocouple
Nuclear - Atomic battery
Sou... | {
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How could Leptoquarks explain Lepton Flavour universality (LFU) violation? I recently read about the possibility of Leptoquarks and that this new particle could also explain a possible LFU violation.
Why would introducing a new particle explain LFU violation?
| What is a leptoquark?
Leptoquarks (LQs) are hypothetical particles that would interact with quarks and leptons. Leptoquarks are color-triplet bosons that carry both lepton and baryon numbers. Their other quantum numbers, like spin, (fractional) electric charge and weak isospin vary among theories. Leptoquarks are enco... | {
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Is pseudo force just an ad hoc number to explain motion in non-inertial frames? Consider an observer in a non-inertial frame $S$ who observes a particle's motion with a relative acceleration $\vec a_s$ and further calculates (or was told by his fellow observer in an inertial frame) the net real forces acting on it as $... | Inertial forces aren't desperate attempts to explain motion in non-inertial frame, but rather correct explanation of motion in non-intertial frames. In physics, everything is about frames of reference.
For observer at inertial frame looking at someone at merry go round, centrifugal force doesn't exist, however this for... | {
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Problem with gravity Sorry if this question is dumb, but I don't seem to have a grasp on it.
Suppose you are on a rock in space, with no external forces acting. The rock attracts you with a force given by
$$F=G\frac{m_1m_2}{r^2},$$
and you also attract the rock with an equal and opposite force. The ground exerts the no... | Just like the ground exerts force on you stopping you from accelerating, your legs push the rock stopping it from accelerating.
| {
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I am moving right and rain is falling vertically down.Why should I hold umbrella at an angle? This is related to relative velocity.I get that, from my moving frame of reference rain is making an angle.But still... it doesn't make sense to hold umbrella at an angle when rain is falling vertically down.
| Imagine yourself in a train car with no roof (to let the rain in) and no windows (so you don't know you're moving). In that car, all you know is that the rain is coming at you at an angle to the vertical. That the rain is falling vertically to someone standing by the train tracks is not relevant to your experience. To ... | {
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Shift Vector in Warp Equation In the Alcubierre metric, why is there a beta with subscript multiplied by a beta with superscript? I know beta with subscript is the shift vector, but what is the difference between the two?
$$\text ds^2 = -(\alpha^2 - \beta_i \beta^i) \text dt^2 + 2 \beta_i \text dx^i \text dt + \gamma_{... | This is how the dot product is defined for covariant and contravariant vectors (without explicitly inserting the metric) i.e, with the metric, the dot product would look like $$\beta \cdot \beta = \beta_i \beta^i = \gamma_{ij} \beta^j \beta^i$$
Note that $$\beta_i = \gamma_{ij} \beta^j $$ and $$\beta^i = \gamma^{ij}\b... | {
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Is temperature of 1 Kelvin equivalent to 1 eV in natural unit? We know that the Boltzmann's constant, $k_B$=8.617 $\times$ $10^{-5}$ eV/K. Now in the natural unit, $k_B=1$.
So can I say, in the natural unit, 1 K temperature is equivalent to 1 eV in energy? 300 K is equivalent to 300 eV?
Or am I missing something?
| No. The situation is similar to the situation in special relativity where we use units in which $c = 1$ instead of $c = 3.00 \times 10^8 \text{ m/s}$. Using $c = 1$ does not mean that 1 meter is now equivalent to 1 second. It means that we are defining our unit for distance to be the distance traveled by light in 1 ... | {
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What is the definition of a magnet or a magnetic field? Electric forces are the forces which come about between two types of charges, positive and negative. Gravitational forces are the forces between matter. Nuclear forces are the forces which act on the atomic scale and are quantum mechanical forces, they act between... | There is a very good explanation of how electric charges in motion create the thing we call a magnetic field in this paper, "Magnetism, Radiation, and Relativity", Supplementary notes for a calculus-based introductory physics course by Daniel V. Schroeder, Weber State University (http://physics.weber.edu/schroeder/
dsc... | {
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What are the physical principles at play when a glass is stuck to a wet table? I've decided to write a relatively detailed paper on the following situation, but I'm finding the topic quite hard-to-google. Imagine a glass table with water spilled onto it. Once a drinking glass is placed on top, it becomes quite difficul... | Let's model the glass as a cylinder with radius 4cm and height 10cm.
Air pressure is about 101,000Pa, there is a downward force due to this pressure on the top of the glass, usually balanced by a similar upward force due to the air inside the glass.
The water makes an airtight seal if the surface is flat enough. When ... | {
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Does inverse square law indicate beam angle of radiation source? I was told that the radiation intensity from a particular X-ray machine falls in half every 4 feet. I think this should tell us something about the angle of the beam from the machine. I use trigonometry to restate the assertion as follows:
Radiation is ... |
the radiation intensity from a particular X-ray machine falls in half every $4$ feet
This is the important part. After $4$ feet you have half the original intensity. After $4$ more feet, you have half of that, or $1/4$ of the original intensity. After $4$ more, $1/8$. And so on.
This is exponential decay
$$I = I_0 \c... | {
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Is there any experiment to check discreteness of space? This article from 2015 seems to suggest that there will be experiments to check discreteness of space:
If space-time is discrete, there should be imperfections. And even if
rare, these imperfections will affect the passage of light through
space. No one has looke... | In the article itself it mentions that there have been previous (failed) efforts to detect discreteness of spacetime:
If true, this would distort images from far-away stellar objects, either by smearing out the image or by tearing apart the arrival times of particles that were emitted simultaneously and would otherwis... | {
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Time evolution in quantum mechanics of states not contained in the Hilbert space Eigenstates of, for example, $\hat p$, are not elements of the standard quantum mechanical Hilbert space, i.e. $\psi(x)=e^{ipx}\notin\mathcal L^2(\Bbb R)$. This prompts the question of - given that after measurement the state of the system... | The issue doesn't come up in practice, because perfect momentum eigenstates are idealizations that don't occur in the real world.
In order to measure a particle's momentum with infinite precision and end up with a perfect plane wave, your measurement apparatus would need to be infinitely spatially large. Any real-world... | {
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Why don't we add atmospheric pressure the same we do pressure from other liquids? If we have two fluids $1$ on top of $2$, I know that the absolute pressure of a fluid $2$ is $p_2 = p_1 + \rho gh$ where $h$ is the height of the second fluid, and $p_1$ is the absolute pressure at the bottom of fluid $1$. In other words,... | We do "add the pressures," but only when the effect is big enough to matter.
The density of air at sea level is about $1.2\,\text{kg}/\text{m}^3$.
So the pressure change in a column of air $1\,\text{m}$ high is about $1.2\times 9.8 \approx 12\,\text{Pa}$.
Compare that with change the atmospheric pressure at sea level o... | {
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Why doesn't the double-slit experiment violate the uncertainty principle? In the double-slit experiment, when an electron reaches the detector after passing through the holes, it has a certain momentum which we can measure with arbitrary accuracy.
From this data, we can calculate what momentum the electron had when it ... |
Where have I gone wrong?
Here, the noted with italics.
when an electron reaches the detector after passing through the holes, it has a certain momentum which we can measure with arbitrary accuracy.
Any measurement means new quantum mechanical interactions, i.e. new wavefunctions and boundary conditions on them. In ... | {
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Functional determinants I wish to know what is the result of this Gaussian Functional Integral
$$Z[\chi] = \int[\mathcal{D}\phi]~e^{-i\int d^dx ~\phi^2\chi}$$ where $\phi, \chi$ are position dependent fields. Now, my question is whether
$$Z[\chi] = (\det\chi)^{-1/2} ?$$
But, since, $\chi$ is not an operator just a scal... | $\det \chi$ is a slight abuse of notation. You are actually computing the determinant of the multiplication operator $m_\chi:\phi \mapsto\chi\phi$. As @CosmasZachos pointed out in the comment, this is the product integral :
$$\det m_\chi = \prod_{x\in\mathbb R^d} \chi(x)^{\text d^dx} = \exp\left(\int_{x\in\mathbb R^d}\... | {
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Why does the electron not spin? The goto answer to that question is that the electron is a pointlike particle and cannot spin.
The electron is not pointlike though. It is described by a wavefunction. One can prepare the wavefunction to describe a very small electron, but not a point-like electron.
Is there a genuine an... | I want to address this:
The electron is not pointlike though. It is described by a wavefunction. One can prepare the wavefunction to describe a very small electron.
The standard model electron is a point particle. The wavefunctions used in the quantum mechanical models to model an electron , call it $Ψ$, which define... | {
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Why aren't all quarks clumped together in one giant hadron? As far as I am aware, the strong interaction is attractive only, and its carrier, the gluon, is massless meaning it has unlimited range. If this is the case, how come we only observe quarks in pairs and triplets? What's preventing every quark in the universe f... | The strong interaction is not "only attractive". A $qgq$-vertex has a color factor associated with it that depends on quark color and the gluon color/anti-color. The total color factor for a 2 vertex diagram is:
$$ C=\frac 1 2 c_1c_2 $$
A positive (negative) color factor is attractive (repulsive).
Interactions are as f... | {
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Antiunitary operators and compatibility with group structure (Wigner's theorem) From Wigner's theorem, we get that a physical symmetry can be described either by a unitary or antiunitary operator, eventually with a phase factor, as in here.
However we have to respect the group structure, so we need to have:
$$O(f)\circ... | Time reversal is a implemented at the level of the Hilbert space $\mathcal H$ by a (projective) representation $\rho$ of $Z_2$, with $\rho(0) = \mathbb I$ and $\rho(1) = T$. Note that
$$\rho(0+0)=\rho(0)=\mathbb I = \mathbb I\circ \mathbb I = \rho(0)\circ \rho(0)$$
$$\rho(0+1)=\rho(1)=T = \mathbb I \circ T = \rho(0)\c... | {
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Why is everything not invisible if 99% space is empty? If every object is $99$% empty space, how is reflection possible? Why doesn't light just pass through?
Also light passes as a straight line, doesn't it? The wave nature doesn't say anything about its motion. Also, does light reflect after striking an electron or at... | The solar system is also mostly empty space. So why doesn't an asteroid from outside the solar system pass straight through? The solar system has many massive objects with wich the asteroid could interact gravitationally. If an asteroid would pass through the solar system it would at least be deflected by the sun a lit... | {
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If air is a bad conductor, how does fire heat up a room? If air is a bad heat conductor, how does fire heat up a room?
Could someone help me, as I really don't get this?
| There are three mechanisms at play: conduction, convection and radiation. Radiation is the most immediate. Your environment irradiates you with black body radiation at room temperature (assuming that you are in a room at 20 C/ 293 K). As soon as your fire burns it emits black body radiation at a temperature of a 600 C ... | {
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A pure rolling sphere problem Question: A sphere $S$ rolls without slipping, moving with a constant speed on a plank $P$. The firction between the upper surface of plank $P$ and the sphere is sufficient to prevent slipping. Friction between plank and ground is negligible. Initially plank $P$, is fixed to ground with he... | It might come as a surprise to you that the frictional force between the sphere and the horizontal plank when the sphere is rolling without slipping along the plank at constant velocity is zero.
This must be so as the only possible horizontal force on the sphere is the force of friction on it due to the plank and this ... | {
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States on finite dimensional Hilbert spaces In my quantum theory lecture we talked about states on finite dimensional Hilbert spaces and had the following statement:
Let $\mathcal{H}$ be a finite dimensional complex Hilbert space, $\mathcal{L}(\mathcal{H})$ the set of bounded linear operators, $\omega \colon \mathcal{L... | Every pure state $\psi\in\cal H$ gives rise to a linear operator $\omega:\cal O\mapsto \omega(\cal O)$ by the familiar $\cal O\mapsto\langle \cal O\psi,\psi\rangle\,.$ For two arbitrary pure states $\psi$ and $\tilde\psi\,,$ there always exists a unitary operator $A$ such that $\tilde\psi=A\psi\,.$ Therefore, restricte... | {
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Blackbird debate: Case when speed of car smaller than speed of downwind I am puzzling my head around the Veritasium famous video right now: https://www.youtube.com/watch?v=yCsgoLc_fzI
Also the Physics Olympiad problem Part B1 https://www.aapt.org/physicsteam/2019/upload/USAPhO-2013-Solutions.pdf
I fully understand (I h... |
Is it possible that the car exceeds the one of the belt
Yes, that is possible. Check out the treadmill videos that they have made, where the small model they built, was able to do this
| {
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What is the size of the magnetosphere of a neutron star? More precisely, what are the factors that influence that size (mass of the star, spin velocity, accretion disk around it, ... do any of those things matter?)
I tried reading this. I cannot: [paper about relative size of neutron stars' magnetospheres]
Is there an ... | The size of a neutron star's magnetosphere (in-so-far as it can be approximated as something spherical) is, to within a small numerical factor, given by the Alfven radius. This is where the magnetic energy density equals the kinetic energy density of the surrounding gas/plasma.
Using Gaussian units, the energy density ... | {
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Why does running spend more energy than walking? The study energy expenditure of walking and running concludes that running spends more energy than walking.
My understanding is that although running makes one feel more tired, that only indicates that the power was higher (since the time of displacement was shorter), bu... | When running, muscles require a higher consumption of oxygen, so it's expected that a significant part of the ATP synthesis takes the fermentation route, which is knowingly less efficient than the usual cellular respiration.
| {
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Does real life have "update lag" for mirrors? This may sound like a ridiculous question, but it struck me as something that might be the case.
Suppose that you have a gigantic mirror mounted at a huge stadium. In front, there's a bunch of people facing the mirror, with a long distance between them and the mirror.
Behin... | Even without a mirror there's a lag, because it takes time for light to go from the man to the audience. Divide the distance from the man to the audience by the speed of light to get the delay.
With the mirror, the light has to flow from the man to the mirror and then reflect to the audience members. Therefore, the tim... | {
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Linear combination of group generators In Matthew Robinson's book Symmetry and the Standard Model he explains that we have generators for rotations $J$ and for boosts $K$. To analyse the group structure, we will look at $N^\pm = J \pm i K$ though and find, that both $N^+$ and $N^-$ form a Lie algebra, which means they ... | Given a Lie group $\mathcal{G}$ the generators of the transformations in the group $\mathcal{G}$, lives in the Lie algebra $\mathfrak{g}$ associated.
A Lie algebra as you can tell, is a particular kind of an algebra. An algebra $A$ over a field $\mathbb{K}$ is a $\mathbb{K}$-vector space with an additional structure, t... | {
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Why is centrifugal force called fictitious? When an object undergoes rotation, from the object's reference frame, which is a non-inertial reference frame, the object feels there is a radially outward force, a centrifugal force, acting on it. However, from an inertial reference frame, this force doesn't exist at all. Th... | According to the basic Newtonian formulation of mechanics, "real" forces come in couples: a force (action) and its reaction acting on the source of the action. Furthermore "real" forces are independent of the used reference frame. Fictitious forces, as centrifugal or Coriolis one, violate both conditions. This is the ... | {
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Why propagator pole is associated to the mass? We say that the pole of the all-orders photon propagator, $$\frac{1}{q^2[1+\Pi(q^2)]}$$
doesn't shift if $\Pi(q^2=0)$ is regular. This amounts to say that the photon remains massless to all orders in perturbation theory. Conversely, the fermion propagator,
$$\frac{i}{\not ... | Here is a heuristic argument.
*
*It's a fact that a full connected propagator/2-point correlation function $\tilde{G}_c$ of a Lorentz-invariant theory tend to have a simple pole $\tilde{G}_c \propto \frac{1}{p^2-m^2}$, where for the sake of the argument $m$ is some constant.
*Given that a correlation function/scatt... | {
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When does a falling flowerpot reach maximum speed? I’m interested to know when a flowerpot of begonias might reach terminal velocity after falling off a high balcony. I’ve read that that a penny reaches terminal velocity after just 15 metres. I think then, it would take longer (farther) for a filled flowerpot but don’t... | This is really a comment, as your question can only be answered with a formula.
the terminal velocity formula,
v = the square root of ((2mg)/(ρAC)).
m = mass of the falling object.
g = the acceleration due to gravity. ...
ρ = the density of the fluid the object is falling through.
A = the projected area of the... | {
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Topologically, is a curvature singularity just a hole? Topologically speaking, a hole can be introduced into a manifold and it will still be a manifold, e.g. remove points within a 2-sphere of some radius from the cartesian plane and you'll still have a manifold.
Penrose's singularity theorems prove the existence (math... |
Or instead, is the singular structure not a part of the manifold, as suggested in @benrg's answer to this question, and thus is not a topological hole?
These are not two different interpretations. A topological hole is precisely something that is not part of the manifold.
Im wondering if, in my naive view, maybe a t... | {
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Proton Electron Merger Can somebody explain what would happen if an electron & a proton, very close to each other are left to "fall" to each other in a straight line?
| There is a probability that they will form a neutron, a hydrogen atom in some s state or an unbound electron proton system. Each of these possibilities can occur with a relative probability depending on the initial state. So it is not correct to say that a hydrogen atom must result, even without specifying in which sta... | {
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Adiabatic evolution of superposition of states I am supposed to find a specific superposition of eigenstates of a time-dependent hamiltonian $H(t)$. The hamiltonian is of the form : $H(t) =\sum_i \left(J\vec{\sigma}_i\cdot \vec{\sigma}_{i+1} + h(t) {\sigma}_i^z \right)$. Periodic Boundary Condition.
I have taken the fo... | Your states have an arbitrary global phase due to $U(1)$ gauge symmetry. To remove it, you need to fix a common gauge across all your states. A convenient way to do this is to iteratively impose parallel transport, i.e. modifying the phase of your $|n_i\rangle$ state such that the condition $\text{Im}\langle n_{i-1}|n_... | {
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True or False: energy is conserved in all collisions Using introductory physics, how would you answer this question? (I have a disagreement with my instructor and I’m curious to hear your input)
One of us says true because the question doesn’t specify “kinetic energy,” or a “system” and all energy is always conserved. ... | Sadly, the correct answer is "Test-setter has failed their practical exam".
The question is worded ambiguously, and any "correct" answer will be determined by psychology not physics - namely what interpretation of the ambiguous question was in the test-setter's mind. There is no way to answer it purely based on coursew... | {
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What is the difference between circular motion and rotation? I've tried so hard to understand the difference, yet no progress. There is a lot of answers here, on Quora, on Youtube,... but everyone give a different answer.
So can you please give a simple yet satisfactory answer?
Someone says that rotation is only about ... | A rotary motion is one in which a body turns around its centre of mass. A purely circular motion is one in which a body follows a circular path but does not turn around its own centre of mass.
If you walk around a circle, you rotate. Although the centre of the circle is your centre of motion, your body also turns aroun... | {
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Cathode Ray Oscilloscope Is Cathode Ray Oscilloscope (CRO) a device that measures and displays using the electron beam from cathode rays? Are cathode rays its working principle? Does it mean that the line or point showed on CRO screen is the result of electron beam from cathode rays strike on the CRO screen?
| Yes, that is true. You can think of a cathode-ray oscilloscope as an AC voltmeter, where you can actually see the waveforms and measure their peak values and study their shapes.
The oscilloscope responds to very high frequencies because the electron beam that "writes" the waveform onto the screen can move very much fas... | {
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What has Euler's number $e$ to do with exponential decay? I know how to derive the formula for "quantity at time $t$" for some decaying materials. You can see the derivation here. But, what I don't get is that what the number $e$ is doing here? We get the value of $e$ from the equation of compound interest. Compound in... | Compound interest and radioactive decay both vary exponentially with time. That simply means that in any set period of time, the value changes by the same fractional amount.
Any exponential function can be written using any base. If the number of remaining atoms in a sample is given by $A * e^{-Bt}$, then it is also g... | {
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Working of liquid lens we were recently introduced to the liquid lens in school, but i havent quite understood the working of it. I realize that the curvature of the lens is changed by some process, but i dont exactly know what, and how it would work. Itd also be great if you could mention what factors would affect the... | Considering the physical principles behind it, a liquid lens is no different from a solid one. The focal length $f$ is affected by the refractive index $n$ of the lens material (assuming a refractive index of 1 in the surrounding medium) and the curvatures $R_1$ and $R_2$ of the two surfaces of the lens, as well as it'... | {
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Why is a liquid-vapour mixture at thermal equilibrium called saturated? If we have a liquid-vapour mixture at thermal equilibrium, apparently this means the mixture is saturated.
How?
From what I understand, saturated means it is on the verge of converting to either liquid or gas. But it also apparently means that the ... | There is some variation in the use of the terms, but in general:
The term "saturated" refers to vapor in equilibrium with liquid at or above the normal boiling point (boiling point at one atmosphere, in the case of water). Adding or removing heat heat increases the vapor component or increases the liquid component, res... | {
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What does the absence of these Goldstone boson interactions mean physically? I have read that in several statistical models exhibiting spontaneous symmetry breaking, the resulting Goldstone bosons do not interact with each other via $\theta^{2n}$ terms — only via derivative terms like $(\nabla\theta)^2$.
For instance, ... | When there's a single $\nabla \theta$ term, this tells you that the theory has a shift symmetry of $\theta \mapsto \theta + a$ which moves you between different vaccua (each with a different $U(1)$ charge). Expressions can become logner with additional Goldstones but the principle is the same. Your expression with $(\t... | {
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Is there cosmological redshift within the Milky Way? Cosmological redshift is based on the idea that the universe is expanding. When the universe doubles in size, or scale factor, the wavelength of light doubles. But the Milky Way is not expanding so my guess is that there is no cosmological redshift within the Milky W... | Even if the Milky Way is expanding with the Hubble flow (and most cosmologists believe that it isn't), the expansion would be difficult to measure.
The size of the Milky Way $d$ is approximately $10^{21} \;\text{m}$.
Using $$v=Hd$$
with Hubble's constant in SI units of about $2\times 10^{-18} \;\text{s}^{-1}$
means tha... | {
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What units is strong nuclear charge measured in? Do particles have a strong nuclear charge in the same way as they have electric charge? If so what unit would be used to measure this? Would it be measured in Coulomb for instance?
| The electromagnetic force couples to electric charge, the gravitational force couples to mass charge and the strong nuclear force couples to color charge. Photons are the exchange bosons in the electromagnetic force, gravitons - if they exist- are the exchange bosons in the gravitational force.
The strong nuclear force... | {
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Why does water contract on melting whereas gold, lead, etc. expand on melting? My book mentions that water contracts on melting, but the book doesn't give any reason why it does so. It is mentioned that:
$1\,\mathrm g$ of ice of volume $1.091\,\mathrm{cm}^3$ at $0^\circ\mathrm C$ contracts on melting to become $1\,\ma... | The melting phase transition transforms the long-range ordered-crystalline solid structure into the short-range-ordered average liquid structure. Looking at the process from the solid side, melting can be seen as the dramatic effect of a collective building-up of defects in the solid over a limited interval of temperat... | {
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Mass of All the Neutrinos I have read that the Sun produces $2 \times 10^{38}$ Neutrinos per second weighing in at approximately 8 MeV.
I have 2 questions.
*
*Is there any way to calculate how many neutrinos have been produced in the past 13.5B years?
*Is this mass part of the calculation of the "visible" mass of t... | To add to ProfRob's excellent answer, an excerpt from Simon D.M. White's 2018 essay, Reconstructing the Universe in a computer: physical understanding in the digital age:
A
possible solution, that dark matter might be made of neutrinos, was greatly encouraged by a
1980 tritium decay experiment which claimed an electro... | {
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Why does twisting a cork make it easier to remove from a bottle? When we want to remove a cork from a bottle first we turn the cork. Turning in one direction makes it easier to remove in the axial direction.
Does anyone know something more about this?
|
Why does twisting a cork make it easier to remove from a bottle?
When you apply an axial force and, possibly, a torque to the cork (and, opposite, to the bottle), you generate a state of tangential stress between the cork and the bottle neck.
With $A$ being the area of contact, $N$ the axial force and $W$ the torque,... | {
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"question_score": "34",
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"answer_id": 7
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Phase transition vs. critical phenomena Just trying to get some clarity in terminology: is phase transitions synonymous with critical phenomena? At the first glance they mean the same thing, but I am not sure whether phase transitions really include such phenomena as Anderson localization and percolation, which are not... | In thermodynamics, phase transition means the transition from one phase (solid, liquid, gas, or other phase) to another phase. Also in thermodynamics, critical point is the transition from where two separate phases exist to where only one phase exists. Beyond the critical point, only one phase exists.
Approaching the c... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why can't the speed of gravitational waves be greater than the speed of light if the universe can expand faster than the speed of light? Since the expansion speed of the universe can be greater than the speed of light, why can't gravitational waves, which also uses space as the medium, travel faster than the speed of l... | While gravitational waves and cosmic expansion are both gravitational phenomena, they are not the same.
The universe "expanding faster than the speed of light" is not a local issue. The Hubble parameter describes the expansion rate now $H_0 \sim 70$ km/s/Mpc (kilometers per second per megaparsec). So for every megapar... | {
"language": "en",
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Why don't we use absolute error while calculating the product of two uncertain quantities? I've found a rule that says, "When two quantities are multiplied, the error in the result is the sum of the relative error in the multipliers."
Here, why can't we use absolute error? And why do we've to add the relative errors? W... | It basically comes from calculus (or more generally just the mathematics of change).
If you have a quantity that is a product $z=x\cdot y$, then the change in this value based on the change of $x$ and $y$ is$^*$ $\Delta z=x\Delta y+y\Delta x$. So then it is straightforward that
$$\frac{\Delta z}{z}=\frac{x\Delta y+y\De... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why do little chips break off so easily from strong neodymium magnets? I have some strong toy neodymium magnets. Typically after a while little chips start breaking off, unlike from most other small metal objects, like in this image.
It could of course be that neodymium is more brittle than metals used for other objec... |
I was wondering if it could have to do with internal tensions that are not present in non-magnetic objects, maybe due to adjacent domains of different magnetization?
That you are right. The point is that the material is not only sintered but this time it is under the influence of a strong external magnetic field whic... | {
"language": "en",
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The faster you move, does it take more and more energy to increase your speed at the same rate? I'd like to confirm this somewhat counterintuitive result. Starting with the definition of kinetic energy:
$$E = \frac{1}{2} mv^2$$
Assume a vacuum, no external forces, and starting from rest. Adding some energy $E$ to the s... | The algebra is correct, but the interpretation may not be what you intend. In your description, there is a contradiction between "no external forces" (conserved momentum) and changing velocity from $v_0$ to $v_1$ (non-conserved momentum).
You refer to a "rocket" that is "adding some energy $E$ to the system by burning ... | {
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Proving that flux contribution due to rotation is zero for a triangular loop welded to infinite wire set up
Consider a triangular loop attached at the vertex to an infinitely long wire which has time varying current flowing in the +x direction
Adapted from JEE advanced paper-1 2016
One may find that the contributio... | I have drawn the set up in four different configurations upon rotation.
The black arrow is the area vector, and x is a point marked inside the triangle. We see that under rotations about the axis, the perpendicular distance from the axis to points inside the triangle remains fixed eg: the distance to x is fixed. This ... | {
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How to understand the ambiguity of vector resolvation? When we solve problems where there is a pendulum suspended using a tight, inextensible string and the question asks about the tension developed in the string at the highest point of the bob's swing. The following is the conventional approach to solve the problem.
... | The mg sin(θ) produces a torque causing an angular acceleration. The T - mg cos(θ) is not zero. It must provide a centripetal acceleration.The mg – T cos(θ) gives a downward component of acceleration.
| {
"language": "en",
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Computing the longitudinal and traceless part of the left hand side of Einstein's equation I am reading a textbook on cosmology. Consider $G^i_j$, the left hand side of Einstein's equation. If $\Psi$ and $\Phi$ are first order perturbations to the time and spatial components respectively of the metric, $G^i_j$ can be w... | We are working in Newton gauge which is also called longitudinal gauge. The perturbed metric can be written as $g_{\mu\nu} = g_{\mu\nu}^{(0)} + g_{\mu\nu}^{(1)}$ with,
\begin{equation}
g_{\mu\nu}^{(1)} = \left(\begin{matrix}
2\psi & v_i \\
v_i & 2\phi \delta_{ij}+h_{ij}
\end{matrix}\right)
\end{equation}
Unlike in syn... | {
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Interesting answer as a range of tension in pulley-block-plane system I want some intuitive understanding on why there will be a range in tension in the below question. (On solving we will get that the system is at rest ($a=0$) and since its starts from rest the blocks will be stationary). Now, this seems experimentall... |
I initially thought friction might change but as the blocks are stationary it's not the case
It's not that friction is changing over time, it's that the specific value for friction (in the static case) is unknown, so the specific value for tension is also unknown. You know the maximum possible value for static frict... | {
"language": "en",
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Can an observer in a double tidal locked system figure out it is orbiting? Tidal locking, when the spin rate of a body matches the orbital rate so that it always faces the other body with the same side, usually occurs for just one of the bodies in orbit. However, there are also systems in which both bodies are tidally ... | There would be Coriolis and centrifugal forces due to the rotation of the reference frame. These could be detected. For example, a Foucault pendulum with its vertical axis perpendicular to the plane of the orbit would precess once per “year”.
| {
"language": "en",
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Can nothing other than light have speed independent of source? Is it impossible for there to be some phenomenon that travels at a different speed than light to have speed independent of the source? Because if there were such a phenomenon there would be competing formulas for time dilation, correct?
To be clear, this is... | All waves in a medium (e.g. sound waves, surface waves in water) travel with a speed that is independent of that of the source (that is ignoring dispersion, which would create a velocity dependence via the Doppler effect)
In a vacuum, it is only light (i.e. electromagnetic waves), gravitational waves, or other massless... | {
"language": "en",
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How can the change of the Earth's temperature be determined with more than 1/10 K accuracy as IPCC suggests? How can one determine the extent of global warming (expressed as a temperature difference) with such precision? The latest IPCC report states temperature changes to fractions of one degree - without actually kno... | Temperatures are recorded at many weather stations in lots of countries.
This data has been recorded accurately for many years, then an average is done over 30 years, for example, to get the yearly change to $0.1K$ accuracy.
| {
"language": "en",
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Torques acting on a unicyclist When a unicyclist rounds a bend, he or she has to tilt in order to generate a frictional force, the friction acting as the centripetal force constraining the unicyclist to a circular path. However, this tilting causes the combined weight of the unicyclist and the unicycle to have a torque... | Why do you need to lean forward when standing on a train when it is accelerating? You displace your CG to be out of line with the normal force on your feet. In that case, the torque about your CG due to the normal force balances that due to the static friction on your feet.
The case of the unicyclist may be similar, th... | {
"language": "en",
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Why is there an $i$ in the definition of hadronic decay constants? The decay of (for example) a pion can be parameterized by a decay constant $f_\pi$ defined via
$$ \langle 0 | \bar d \gamma_\mu \gamma^5 u |\pi^+(p) \rangle = i f_\pi p_\mu $$ $$ f_\pi \approx 131 \text{ MeV.}$$
My question is why do we include the $i$ ... | Well, you may define your conventions, and physicists are perverse enough to actually do that (watch them...), any way you want. This is the dominant convention, and the logical "chain of custody", so to speak, is, schematically,
$$
\pi^+\mapsto \pi^+ + f_\pi \theta^+, \qquad \leadsto \\
J_{5}^{\mu~~+} = \frac{δ{\mathc... | {
"language": "en",
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What are quantum fields made up of? If quantum fields are mathematical entities made up to explain nature, what they explain is definitely something physical and is made up of something. So why can’t there be an answer to what these mathematical quantum fields are made up of?
I mean, if physicists are making them up, t... | I'm going to give a bit more literal of an answer with a bit less story since that's my taste. What we call a quantum field is an operator as a function of position and/or time, i.e. at every $x, t$ there is an operator $\hat{\phi}$. Just like other operators, this operator acts on elements of the hilbert space. In mat... | {
"language": "en",
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Why does a fluid follow the wing? When air is moving above and under a wing that is curved, why does the air at the top of the wing follow the wings shape and go downwards when it could just go in a straight line? It doesn't make sense to me.
| When air flows over a wing (or more generally, when any fluid flows over a solid surface), viscosity tends to make the air stick to the surface and creates a region of sheared flow (the boundary layer) where the speed varies between zero at the surface and a larger value farther out in the flow. Assuming sufficiently l... | {
"language": "en",
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If the experimentalist does not look at the recorded data in the 'which-way' experiment, will it affect the inference? Related question quantum eraser question
In https://en.wikipedia.org/wiki/Double-slit_experiment#%22Which-way%22_experiments_and_the_principle_of_complementarity
"Which-way" experiments and the princi... | It is enough to put detectors on the slits to measure which way was taken. The interference pattern will dissapear. You don't need a human being to notice the results. The point is interaction (an event happening somewhere on slit A or B, not that someone notices it). Things happen without someone spending attention to... | {
"language": "en",
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Index Notation Question We're busy doing a GR course, and index notation has always been something that confuses me. In particular, is there a difference between the following, and if so, what is it?
$A^\mu_\nu$; $A^\mu{}_\nu$ and $A_\nu{}^\mu$
| Let's say we start with a general (not symmetric or anti-symmetric) tensor $A_{\mu\nu}$. Note $A_{01} \neq A_{10}$ (and similarly for other combinations). Then
\begin{equation}
A^\mu{}_\nu = g^{\mu\rho} A_{\rho\nu} \neq g^{\mu\rho}A_{\nu\rho} = A_\nu{}^\mu
\end{equation}
which you could see by explicitly writing out b... | {
"language": "en",
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Poynting Vector Perpendicular to Surface In Spherical coordinates if we have a source at the origin generating $E$ in the $\hat{r}$ direction and $H$ in the $\hat{\phi}$ direction then our Poynting Vector will be in the $\hat{\theta}$ direction. When considering the power of such a source we know that the total power i... | It's possible to pose questions about field configurations that can't be made! An everywhere radial $\mathbf E$ field suggests a net charge with a spherically-symmetrical distribution, which is fine. But an everywhere azimuthal $\mathbf H$ field suggests a current in the $z$-direction, which is inconsistent with a sphe... | {
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Why is Hamilton's principle (or principle of least action) still valid in a relativistic field theory? I am struggling to understand why the principle of least action which is derived in classical mechanics from d'Alembert's principle continues to be valid in a regime that treats a relativistic field. What is it that t... | Here is one possible line of reasoning:
*
*In the case of relativistic point mechanics, one can still use d'Alembert's principle and the relativistic version of Newton's laws to derive Lagrange equations. (The main difference compared to the non-relativistic case is the form of the kinetic term in the Lagrangian.)
... | {
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Is space really expanding? In a book called "Einstein, Relativity and Absolute Simultaneity" there was this sentence by Smith:
There is no observational evidence for a space expansion hypothesis. What is observed are superclusters of clusters of galaxies receding from each other with a velocity that is proportional to... | Smith is just wrong.
He is right to say that there is no direct observational evidence (in a naive sense) that space is expanding rather than space being fixed while galaxies spread apart through it. They both just look like galaxies getting farther apart.
But there is a lot of observational evidence that special and ... | {
"language": "en",
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Does electron physically move in an interband transition? How does electron move say from ground state energy level to first excited state? Is there any actual displacement in terms of motion?Is there a way to logically think about this by the help of creation and annihilation operators?
I know this is a quantum mechan... | The bands are the many-to-one relation between energy and momentum for electrons in a crystal. This relation is not position dependent. (I am not talking about band warping near crystal borders.) To go from one band to another just means that an electron's energy, and possibly momentum, changes. For a so-called vertica... | {
"language": "en",
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Is it true that Maxwell equations are interpreted by taking right side of formula as the "origin" and the left part as "consequence"? When books or various references interpret the meaning of Maxwell equations, they typically state that the source (origin of the phenomena) is the right part of the formula, and the resu... | You are correct. Maxwell-Faraday states that $\vec{\nabla} \times \vec{E}$ is the same thing as $-\frac{\partial \vec{B}}{\partial t}$. Both quantities express the same phenomenon. There is no cause and effect in either direction.
Once you express the fields in terms of derivatives of the vector potential the two expre... | {
"language": "en",
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Rabi Hamiltonian with three instead of two Pauli matrices This question was motivated by a question on Mathoverflow, in which a Hamiltonian is considered that looks like the Rabi Hamiltonian, but with three instead of two Pauli matrices:
$$H=\omega a^\dagger a+\Delta\sigma_z+g_x\sigma_x(a+a^\dagger)+ig_y\sigma_y(a-a^\d... | I probably should have searched more extensively before posting here, I did eventually find this Hamiltonian in Exceptional and regular spectra of a generalized Rabi model by Michael Tomka, Omar El Araby, Mikhail Pletyukhov, and Vladimir Gritsev. The context is different from quantum optics (which is probably why I cou... | {
"language": "en",
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Why the mechanism of everything in the universe has a pattern? why everything in the universe has a pattern which can be identified and understood to determine outcomes, properties, effects of almost everything. I am saying that couldn't the universe be like patternless, non-deterministic and chaotic. For example why t... | Humans like looking for patterns in nature and when we find them we show each other in books, lectures etc...
It is remarkable that so many patterns have been discovered.
However there are lots of things where patterns haven't been found, we probably don't talk about those as often, especially on 'Physics StackExchange... | {
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Will two bodies initially connected to and revolving around each other, start spinning when disconnected? Two extended bodies are connected with a string and revolve around each other (that is, around the center of mass of this system). No gravity, no external forces.
The string is cut, and they start to depart from ea... | Take a ball of radius $r$ connected to a thin center-post by string of length $R$. The post accelerates up to and stays at a high final rotation speed $\Omega$. Obviously the ball revolves at $\Omega$. Now imagine case 2 where a thin, light ball-bearing is connected to the string and the ball rotates within it.
Lookin ... | {
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Pressure in open tube
In the L shape tube shown in the image shouldn't the pressure at all points A, B,C,D be same as atmospheric pressure?
I have to find relation between A, B, C and D
| The "$\omega$" in the drawing suggests that the tube is rotating. In that case there will be a pressure gradient in the horizontal part to provide the centripetal acceleration:
$$
\rho_{\rm air}\omega^2 r = \frac{dP}{dr}
$$
| {
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Issue in deriving Ehrenfest's theorem Working in Schrodinger picture, while deriving Ehrenfest's theorem, we go -
$$
\frac{d}{d t}\langle A\rangle=\frac{d}{d t}\langle\psi|\hat{A}| \psi\rangle
$$
$A$ is an operator. Expanding RHS-
$$
\frac{d}{d t}\langle A\rangle=\left\langle\frac{d}{d t} \psi|\hat{A}| \psi\right\rangl... | We us the partial derivative because there are other variables in play --- such as $x$ and $p$, both of which may be time dependent. The partial derivative symbol is used because it implies that we are keeping all the other variables fixed when we vary $t$.
Using the "$d$" derivative would imply that
$$
\frac{d}{d... | {
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The relationship between velocity of centre of mass and angular velocity of a rigid body Consider a rotating object with mass $m$, moment of inertia $I$, along an inclined plane of vertical height $h$. Then simply speaking the following conservation law holds.
$$\frac{1}{2}(mv_{CM}^2+I\omega^2) = mgh$$
Recognise that a... | There are a unique set of $\omega$ and $v_{cm}$.
They are in fact both related to each other by the equation $$\omega=\frac{v_{cm}}{r}$$
| {
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Optimizing for "sudden" change in angular momentum I'm trying to create an elaborate impossible-to-juggle juggling club, by triggering a sudden change in angular momentum via a mechanism inside of the juggling club while it is in mid-air. The mechanism is going to be a torsion spring attached to a flywheel. For the pro... | For maximum effect you will want to maximize the mass and radius of the flywheel and the torque exerted by the spring. You will need to decide whether you want the flywheel to be set spinning or allowed to oscillate. Its axis should be perpendicular to the long axis of the club. Your biggest problem will building a m... | {
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Is the Lagrangian formulation a mathematical inevitability? An analogy with functions:
Say, we have a function $f(x)$ and we have an equation to solve, $f(x)=0$. We can always re-formulate the problem of solving $f(x)=0$ with the problem of extremising $F(x)$, where $F(x)$ is the anti-derivative of $f(x)$. This is pure... | The cheapest way to make $F(x,\,\dot{x})-\ddot{x}=0$ an ELE is with an auxiliary variable $y$ viz. $L=y(F(x,\,\dot{x})-\ddot{x})$, so $y$ is dynamical. You may prefer to add a total derivative $\frac{d}{dt}(y\dot{x})$ to get an alternative Lagrangian, $L=yF(x,\,\dot{x})+\dot{y}\dot{x}$. If $x$ were complex-valued, the ... | {
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Diffraction conundrum Take any macroscopic crystal of VIS transparent monocrystalline type of material like table salt ($\text{NaCl}$), sugar (sucrose), quartz (silica), calcite ($\text{CaCO}_3$), glass (which is non-crystallin, of course) etc etc. A VIS laser beam travels through these crystals effortlessly.
But grind... | The reflectivity, in the visible range, of your samples is nonzero. For large crystals, the light (generally) only passes thru two interfaces, i.e. air-crystal and crystal -air , and thus rather little signal is lost.
WHen you grind it up, not only do the crystal faces of each particle point in random directions, but t... | {
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How is Lorentz force frame-independent? I have studied that the net force on a charged object moving with velocity $v$ under both electric and magnetic fields in given as $\vec{F}=q(\vec{E}+\vec{v}\times \vec{B})$. I have also been told by my teachers that the net force given by this equation is frame independent for i... | Good question. You have stumbled upon the fact that Maxwell's equations are not invariant under a Galilean Transformation. Galilean Relativity says that two objects separating with a relative velocity $\vec v$ will calculate the same force on a particle: $$x'=x-vt$$ therefore $\ddot x'=\ddot x$
So first of all let us m... | {
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"url": "https://physics.stackexchange.com/questions/661883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
Spontaneous emission laser Why exactly can we not create a 'laser beam' from a medium undergoing spontaneous emission inside a cavity?
It seems like the dielectric mirrors are going to cause a standing wave to appear irrespective of whether the medium is doing stimulated or spontaneous emission, and that will produce s... | For driven harmonic motion, there must be a resonant frequency and an oscillating driver, and they must be somewhat close in frequency. You may think of a pendulum for example and not immediately realize we do have that situation in a way. One way to think of it is that driven oscillation situation is mimicked. We hav... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/661983",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
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