Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Can low lying clouds increase the distance of the radar reach from 18 to 120 km? I have seen an article (sorry its Russian) claiming that low lying clouds have increased the distance of the radar reach from the normal 18 km till as far as 120 km. This allowed the military ship to be detected and sunk.
The source claims... | Clouds? Probably not. Temperature inversion? Very possible. This is a well-known effect: atmospheric ducting.
| {
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Dimensional Analysis Does Not Check Out I'm looking over the lecture notes found here, and if you scroll down to the end of page 2, the writers make the statement $ A(\omega _d) = \dfrac{f_0}{\sqrt{\omega_0^2 - \omega_d^2 + \omega_d^2\Gamma^2}}$ The math that leads to this point checks out, but unfortunately the units ... | It's likely there is a small error in the notes.
From context, it seems to me the equation should be:
$$A(\omega _d) = \dfrac{f_0}{\sqrt{(\omega_0^2 - \omega_d^2)^2 + \omega_d^2\Gamma^2}}$$
It is a great skill to be able to sanity-check expressions using dimensional analysis, as you've done here.
| {
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Do we need a quantum gravity theory to model an hydrogen atom on earth? The hydrogen atom is a quantum mechanical system. However, it is also attracted by the gravitational pull of the earth. Therefore, do we need quantum gravity to model its behavior correctly? Conversely, can we study hydrogen atoms on earth to obtai... | Physics is all about formulating what exact aspect of a given phenomenon you would like to describe/understand, and then introducing a minimal model, which is just complex enough to mimic the desired behavior.
For instance, if you encounter pianos falling through roofs, and you want to have a description of pianos spec... | {
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Why is the stress on a body not a vector? In my textbook, Physics, Part II—Textbook for Class XI, there's a line which talks about why stress is not a vector:
Stress is not a vector quantity since, unlike a force, stress cannot be assigned a specific direction. Force acting on the portion of a body on a specified side... | Stress is not a vector because it needs more information than what a vector can provide.
Stress can be represented as a $2^{nd}$ order tensor quantity $\mathbb{T}$ that relates two vectors, the unit normal vector, $\mathbf{\hat{n}}$, of a surface passing through a chosen point and the force per unit surface acting on i... | {
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Ultrastatic spacetime and cosmological constant A spacetime $(\mathcal{M},g)$ is called "ultrastatic", if it admits a set of coordinates such that
$$g=-\mathrm{d}t^{2}+h$$
where $h$ is a Riemannian metric, which does not depend on time. Does there exists an example of a spacetime, which solves Einstein field equations ... | There are none. According to the paper "Ultrastatic spacetimes" by Sebastiano Sonego, for any ultrastatic spacetime,
$$Λ=\frac12 κ (T_{ab} η^a η^b + T_{ab} h^{ab}) \tag{2.6}$$
(where $η$ is the Killing field), so $Λ=0$ if $T=0$.
More informally, the integral curves of the Killing field are geodesics in an ultrastatic s... | {
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What would a standing wave of light look like? I want to know what a standing wave of light would like and what properties it might have that are interesting.
| A standing wave is formed when two (plane) waves of the same frequency travel in opposite directions. If you stand inside a plane wave of light, you see it as a "star": a point of light infinitely far away in the direction the wave is coming from. So if you filled a room up with an optical standing wave (not sure how y... | {
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Contravariant Components (Susskind's book) In his book about SR & classical field theory, Susskind generalizes from the differential of $X'$ (function differential) to any 4-vector. I got stuck there trying to figure out why it is possible and why this for holds for any general 4-vector?
$$ d(X')^\mu = \frac{\partial(... |
I got stuck there trying to figure out why it is possible and why this for holds for any general 4-vector?
In this context $(4.19)$ is the definition of a $4$-vector: An assignment of $4$-tuple to each chart such that the coordinates w.r.t to two different charts are related through $(4.19)$. It is possible to define... | {
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Why do we choose the Dirac delta function as the eigenstate of position operator? When we try to find the eigenstates of the position operator, we get that the product of (x-y) and the eigenstate must be zero. It is obvious then that for x different than y, the eigenstate must be zero.
Now for x equal to y, how do we k... | The position space representation of the position operator $\hat{X}$ is just the variable $x$ so the eigenfunction equation looks like $$x\psi(x)=x_0 \psi(x)$$ where $x_0$ is just a constant. The only (generalized) function that satisfies this relation is $\delta(x-x_0)$.
| {
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Why does a chimney sometimes create a "draft"? Sometimes, while lighting a fire in a wood stove in a basement, the smoke does not exit through the chimney like it normally does. Rather, a large amount of the smoke seems to get "pushed back" into the room instead of exiting through the chimney.
Why does this occur?
| Could be one of two things:
*
*Cool air being forced down the chimney taking smoke with it -
that is, we know that cold air sinks (and hot air rises), and so a column of cold air sinking can force smoke back down into the room.
*It could also be simply a matter of winds blowing downward pushing smoke back inside.
| {
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How can I prove that the chronological (=time-ordered) product of two scalar fields is Lorentz invariant? In my course of introduction to QFT, one of the questions is prove that the chronological product (time-ordered product) of two scalar fields is Lorentz invariant and I'm having trouble coming up with an answer. T... | Operators commute if they are spacelike separated, so that if $(x-y)^2 > 0$, then their ordering is completely irrelevant and the time-ordering operator does absolutely nothing to the product.
If they are timelike separated, then their causal relationship is Lorentz invariant. More mathematically, if $(x-y)^2 < 0$ then... | {
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Is household alluminium foil good for observing photoelectric effect? I have built an electroscope(very primitive, just a wire and some aluminum foil). I'm thinking about using it to observe the photoelectric effect. But I don't have any high frequency wave producing equipment.
So, before I try to obtain something for ... | You can use aluminium foil if you just want to demonstrate the photoelectric effect, without serious measurements. Remove the oxide layer from the aluminium foil with a clean piece of sandpaper, it remains usable for half an hour or so. Then apply the sandpaper again. A cheap sterilization box, easily available online,... | {
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How can storing a bit of information require less mass than an electron? The prefixes ronna-, ronto-, quetta- and quecto- were adopted recently to mean $10^{27}, 10^{-27}, 10^{30}$ and $10^{-30}.$ According to The Guardian, "An electron weighs about a rontogram, and a single bit of data stored on a mobile phone adds ab... | It's stored in the energy of an electrostatic field, which has mass. It's closely related to photon mass, but not exactly the same. Whether a photon has mass or not depends on just how you define/measure it. A resonant cavity holding a photon has more mass than one without a photon in it.
| {
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The relation of time-evolution operators from Schwartz's textbook In the section 7.2.2 of Schwartz's QFT textbook, it says:
define the generation definition of time-evolution operators:
$$U_{21}\equiv U(t_2,t_1)=T{\exp[-i\int^{t_2}_{t_1} dt'V_I(t')]}\tag{7.46}$$
where $V_I$ is the interaction part of Hamiltonian in the... | Schwartz forgot to mention that the time-evolution operator $U(t_2,t_1)$ is antitime-ordered for $t_2<t_1$, cf. e.g. my Phys.SE answer here. Then eq. (7.47) becomes essentially just a telescoping product when discretized.
| {
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How do atoms change their phase randomly to produce incoherent light? From research I found that for one light wave to be coherent, its sources (atoms in a regular light source) need to have the same wavelenght and be in constant phase difference. I can picture why the light has to be monochromatic, because if not, no ... | It is useful to distinguish between EM radiation and an EM wave.
Radiation occurs when excited subatomic particles in the atom return to a lower state. Photons are emitted in the process. In a thermal radiation, the particles are chaotically on the most different energy levels, the photons of the radiation have the mos... | {
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Doubt regarding the mathematics in reflecting surfaces So the question goes as follows
"A reflecting surface is represented by the equation x² + y² =a². A ray travelling in negative x-direction is directed towards positive y-direction after reflection from the surface at point P. Then co-ordinates of point P are"
The... | No, it is ambiguous because a was merely described as being part of an equation.
Had the question said "radius a", that would be what constrains a to what "radius" means -- a radius is always positive.
| {
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Does proper time of a test particle depend on the density of the body, around which the test particle is moving? Does the proper time of a test particle depend on the density of the body, around which the test particle is moving? Say the proper time on earth is not the same as the proper time on black hole, since black... | In some sense, you can say that it does depend; But actually, it doesn't in general. Truly speaking, proper time is a geometrical concept. So from that point of view, it may seemed to be independent of any physical parameter.
But if you take Einstein's field equation into consideration, then the proper time should depe... | {
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How can rigid body have more than 1 speed? I have a doubt that circular ring is a rigid object and we know every point on the rigid obj must have same velocity or speed. But in case of pure rolling of sphere or others we can find topmost point has velocity 2v lowermost have 0 how's that possible..i am not able to visua... | A rotating rigid body has a different velocity (speed + direction) at each point on the body. Only along the rotation axis, the velocity is the same.
This is often described as a velocity field, with the interpretation of velocity (vector) only being a position function.
In the potato example above the body is rotatin... | {
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Need help about motion So,In my book it is written that -
Translational motion is the motion when all the particles of a moving body move the same distance in the same same in the same direction.It is of two types-1)Rectilinear and 2)Curvilinear
Now, my question is that to have translational motion doesn't need to have... | Translational motion is when there is no relative motion between any two particles within the body. When there is any rotational motion of an object about an axis, the particles not on that axis will move perpendicular to their position vector relative to the particles on that axis (circular motion). In case of rolling... | {
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Why doesn't a frothing drink overflow the glass? When you pour an aerated (fizzy) drink such as lemonade or beer into a glass, the froth rises above the liquid level, but usually stops rising when it reaches the rim of the glass, even though you keep pouring and the liquid level continues to rise.
Obviously if you over... | Champagne appears to more carbonated than most other drinks. That would easily why champagne is more "lively". Never heard of any "corking"-type behaviour on other drinks so I would think champagne is more carbonated than most.
Beer = 2 to 4 volumes, or 2 to 6 g/L.
Seltzer = Around 4 volumes, or 6 g/L.
Champagne =... | {
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Why is Newton's second law seemingly not applicable to a ball rolling down incline plane? A homework problem asked us to find the acceleration of a ball (pure) rolling down an incline plane without friction. I thought it was simply $a=g \ \sin(\alpha)$ where $a$ is the acceleration of the CM and $\alpha$ is the angle o... | The key word "pure rolling" means rolling without slipping. In other words, at all times the velocity of the material of the ball at the point of contact with the plane is 0. There is no slipping between the surface of the ball and the surface of the plane.
However, this "pure rolling" condition requires friction. You ... | {
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Remote mass detection and location Could a collection of gravitational sensors (eg. MEMS accelerometers), placed in a 3d grid (eg. 10 x 10 x 10 sensors forming a 1 x 1 x 1 m cube) detect the the presence and position of a remote mass concentration, eg. an airplane in the sky?
What accuracy would be needed to do so?
| Your accelerometers would be measuring acoustic waves from the airplanes and the phenomena. There is already a global array of infrasound detectors and seismometers to monitor for nuclear tests, and their sensitivity is exceptional enough to detect things like plane crashes and probably even cruising planes and trucks... | {
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Why do we use $P_{\rm ext}$ in the formula of $\int p \, dv $ work? To what I can understand-in a piston cylinder arrangement, piston moves out due to net force experienced by piston due to difference between internal pressure of system and external pressure. So, the formula of work done by system on piston, should be
... | The net work on the piston includes all external forces; your equation is correct if there is no gravity and no friction. The net work on the piston is the change in kinetic energy of the piston.
If you want the work done by the system on the surroundings and you include the piston as part of the system, that work is ... | {
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How to prove $\mathrm{Tr}[(\partial_\mu U)U^\dagger]=0$? I am studying ChPT by referring to "A Primer for Chiral Perturbation Theory" by Stefan Scherer.
I'm having a problem with the consideration of terms that appear in the Lagrangian.
The textbook says only $\mathrm{Tr}(\partial^\mu U\partial_\mu U^\dagger)$ is impo... | I should have checked the answer is attached at the end of the book...
The important things are the hint $[\phi, U^\dagger]=0$ and the cyclic property of trace.
$U$ and $U^\dagger$ can be rearranged so that they are next to each other like a c-number in the trace even though $\phi$ and $\partial_\mu \phi$ do not commut... | {
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Is an electron in an hydrogen atom being measured by the nucleus? In an hydrogen atom, the electron interacts with the nucleus by multiple forces, for example the Coulomb force. Does that mean that the nucleus makes quantum measurements of the electron?
EDIT: I became aware that the word ‘measurement’ is not present in... | No. Effects of interaction are described by the Hamiltonian. Quantum measurements are not, they require some "projection", or "collapse" or "choice by hand".
| {
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The motion of a rigid body Consider a rigid body with $n$ forces acting on it. What I intend to know is how to determine the motion of the body, more specifically:
*
*How to determine a point through which the axis of rotation of the body passes
*How to determine the direction in which the axis of rotation points
*... | The general 3D motion of rigid body is not simple; for example, inertia is a tensor. Euler developed the general solution. Rotation in a plane where inertia is a scalar, is a simple special case addressed in most general physics books such as one by Halliday and Resnick.
The general case is developed in Goldstein, Cla... | {
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Why does the attenuation constant not affect the wavelength? Reference: https://en.wikipedia.org/wiki/Propagation_constant#Phase_constant
Let's say I have a propagation constant ($\gamma$), where $jk_z = \gamma = \alpha +j \beta$. Why does the attenuation constant have no role in determining the wavelength, especially ... | I think your question should be asked in a different way, and the answer to that simpler question will also answer your question implicitly, namely: what is the frequency of a damped oscillation, such as $f(t)=e^{-\alpha t } sin(\omega_0 t)$?
First of all, this $f(t)$ is an exponentially unbounded signal and has no Fou... | {
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Do objects always speed up as they fall? In the thermodynamics video I watched, it was stated that objects at higher elevations have more energy. And that objects must speed up as they fall. My question is, "what happens if an object reaches it's terminal velocity? It's still falling but no longer speeds up" Doesn't t... | All physical laws are simplified descriptions of reality. How much simplification is too much? How much error can you tolerate?
If you want to know how long it takes a cobble stone to fall from a second story window to the ground below, you can ignore the air resistance. It won't make enough of a difference for you to ... | {
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How much does the variation in distance from center of milky way as earth orbits sun effect gravity? As the earth rotates around the Sun our distance from the center of the galaxy is varying - what sort of variations does this cause in the force of gravity here on earth?
[Edit]
Given that Earth orbits around the Sun at... | None that we could measure.
The space station orbits the earth. Astronauts are weightless inside. If the space station was far from Earth orbiting the galaxy on its own, it would be the same.
On Earth, there is an effect from the gravity of the Moon. One side of the Earth is closer to the moon. Gravity from the moon is... | {
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What were the first particles, according to different models? At what point, and in what cosmological epoch, did particles begin to exist? What particles were they?
My understanding is that things as we know them involved the Big Bang singularity, and that the particles we currently have evolved from that through a ser... | You are asking about modeling the history of the universe.
At what point, and in what cosmological epoch, did particles begin to exist? What particles were the
The particles existed as far as the theory goes , even before symmetry breaking.
Let us take the Big bang model as it has developed up to now.
If you go to t... | {
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If $I$ is the intensity of the principal maximum in the single slit diffraction pattern, then what will be its intensity if the slit width is doubled? I've found a few results on the internet for this question, all with different explanations or different answers. Some say it'll become 4I and some that it'll not change... | There are three regimes. Given slit width $a$, distance to the screen $d$ and wavelength $\lambda$, if $d<<a^2/\lambda$, the projected slit image on the screen just has the slit width, and the intensity of that image is simply the intensity of the illumination of the slit, independent of the slit width.
Otherwise, in t... | {
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Is Intensive property/Intensive Property an Extensive Property? We know that Extensive property/Extensive Property is Intensive is most of the cases, but is Intensive/Intensive an Extensive property ? if so, is there any examples
| See the other answer on intensive property
Let's think about functions.
Take two functions $f(x)$ and $g(x)$, which are intensive. Hence $f(x+x) = f(x)$ and $g(x+x) = g(x)$
And let's take fraction of them $$\frac{f(x)}{g(x)}$$
What you are asking is if I am changing to $x+x$, how will my $\frac{f(x+x)}{g(x+x)}$ change?... | {
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Trace and index manipulation Imagine that I have a quantity $F_{ab}$ multiplying the stress tensor $T^{ab}$:
\begin{equation}
F_{ab} T^{ab}.
\end{equation}
There is also a metric, say $h_{ab}$. If I want to write the above equation in terms of the trace, $T^a_a = T^{ab}h_{ab}$ (I think this is the right definition). Th... | Simply to find out the trace $T^{ab}h_{ab}$ works and gives a scalar (easy to visualize when the metric is diagonal).
To get the RHS, try $F_{ab}T^{ab}h_{bc}h^{cd}\delta_d^b$=$F_{ab}T^a_ch^{bc}$.
Now do not rewrite the dummy $a$ to $c$ since new $c$ will be different than the esisting one, it becomes $F_{c'b}T^{c'}_ch^... | {
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Under a uniform time dependent magenetic field, which Amperian loop shall we choose to determine the induced electric field? I was reading the Griffin's Electrodynamics textbook and I came up with this question:
Suppose we have a uniform time-dependent magnetic field, I know that by Faraday's Law, we can calculate the... | Sorry this is not the answer but comment field is very limited then I write here.
If there is no wire circle in the uniformly varying magnetic B-fields, there are B-fields and electric E-fields (emf) in space.
Although the B-fields are uniform over the space positions, the emf E-fields become some non-uniform distribut... | {
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"url": "https://physics.stackexchange.com/questions/746835",
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What is the relevance of the Lorentz factor in general relativity? The Lorentz factor is ubiquitous in Special Relativity and is used to express "how much the measurements of time, length, and other physical properties change for an object while that object is moving."
However, in my fairly introductory study of Genera... | $γ$ is a geometrical factor that would appear in Euclidean geometry also if it were taught in a slightly different way. A transformation between two Cartesian coordinate systems with a common origin can be written $\begin{pmatrix} \cos θ & \sin θ \\ -\sin θ & \cos θ \end{pmatrix}$ where $θ$ is the angle between them. I... | {
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How is the slit material not considered an observer? As far as I can tell, the essential process of "observation" is that there is an interaction with something else, providing a means by which any part of the universe noticed something about the thing being observed.
In the double-slit experiment it is absolutely obvi... | If the interaction with the electron changes the state of the system representing the double slit in such a way that you could (in principle) find out through which of the slits the electron had passed, the interference pattern gets indeed lost. Otherwise you will observe interference. You find a nice elementary discu... | {
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Does a double star system have more mass than its constituents? According to Einstein, energy is equal to mass. Consider a planet that is in gravitational attraction to two stars. Normally I would say that the gravitational attraction is proportional to the masses of the two stars. But if they are orbiting each other, ... | Your question is not really different from a proton and an electron joining together to form a hydrogen atom.
When such a combination takes place approximately $13.6\,\rm eV$ of energy is released and the mass of a hydrogen atom is less by the mass equivalent of $13.6\,\rm eV$ $(E=mc^2)$ than the combined mass of an is... | {
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How flat is water? For the purpose of this question, let us define “flat” as meaning “having a surface which is a plane”.
Clearly, the Earth being round, water is not flat. If you take a sheet of water of length $2l$, the middle of it will bulge above a straight line joining the two ends, the height of the bulge being ... | One answer to the question "to what extent are these bulges observable in real life" is the well-known observation that ships at sea, when seen from an observer far enough away (on a clear day), seem to be half-submerged in the water. Also if you are approaching land then (again on a clear day) you will see the mountai... | {
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Asymmetrical meniscus in certain glassware I work in a lab that does tests on water samples, and one test we do measures COD or Chemical Oxygen Demand. For quality control, we have a COD standard which contains a certain amount of Potassium Hydrogen Phthalate that consumes oxygen at a known rate to standardize our meas... | That suggests a patchy adsorbed layer of something foreign on the glass surface itself. I recommend you do a chromic acid dip & rinse, dry, & retest.
| {
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Upper index covariant derivative $\nabla^\mu$ In the book Cosmology by Daniel Baumann, the author states that $\nabla^\mu g_{\mu\nu}=0$, where $g_{\mu\nu}$ is the metric tensor considered (usually the one associated to the Minkowski metric or to the FRW metric), and this is used to reason that we can introduce the cosm... | In General Relativity, the covariant derivative is defined in such a way that $\nabla_\rho g_{\mu\nu} = 0$. This can be seen by writing the covariant derivative explicitly in terms of Christoffel symbols, for example. One may then notice that $\nabla^\mu g_{\mu\nu} = g^{\mu\rho} \nabla_{\rho} g_{\mu\nu} = 0$.
| {
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What happens inside a conductor when the current faces resistance? According to my book resistance depends on the the number of collisions free electrons suffer in any conducting device. Now insulators have no free electrons so how do they have high resistance like what collisions are taking place here? Am I getting re... | I like to think of resistance more as the resistance of current flow. As my resistance becomes very large, less current will flow through my circuit.
You could even think about it with Ohm's law of course. Write it as a function of current vs resistance (I = V/R), if I have a circuit with a set voltage, I could add a r... | {
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Understanding how to terminate recurrence relations in quantum SHO In the coordinate representation solution to the quantum SHO (the solution via differential equations rather than Dirac's "trick") we ultimately work out that our eigenfunction solutions are of the form (up to some constants)
$$u(q) = H(q)e^{-\frac{1}{2... | The reason you state for why we must pick one or the other as 0 is completely true. We can only have one value of $\lambda$ (corresponding to our energy level) and for a given $\lambda$ only one of the even or odd series terminates. The other would continue on as you said but we necessitate that our solution must be no... | {
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Forces that are invariant under Galilean spacetime rescaling $\mathbf x' = \lambda \mathbf x$, $t' = \lambda^2 t$ Consider a force of the form
$$
m \ddot{\mathbf x}(t) = -k\frac{\mathbf x(t) - \mathbf x_0}{|\mathbf x(t) - \mathbf x_0|^d}.
$$
For what values of $d$ is this force invariant under the Galilean spacetime re... | The force is:
$$F=-k\frac{x(t)-x_0}{|x(t)-x(0)|^d}\tag 1$$
Invariant mean that the transformed force has this "structure"
$$F'=-k\frac{x'(t')-x'_0}{|x'(t)-x'(0)|^d}\tag 2$$
with
$$x(t)\mapsto \frac 1\lambda\,x'(t/\lambda^2)\quad,t'=t/\lambda^2 $$
in Eq. (1):
$$F'=-\frac{k}{\lambda}\frac{\left((x'(t')-x'(0)\right)}
{\le... | {
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Why should bulk modulus always be positive?
The minus sign that appears in Equation 12.39 is for consistency, to ensure that $B$ is a positive quantity. Note that the minus sign ($–$) is necessary because an increase $\Delta p$ in pressure (a positive quantity) always causes a decrease $\Delta V$ in volume, and decrea... | As requested, a simpler version of the excellent answer by Buzz: If you pulled on this hypothetical material, it would shrink. We just don’t see that behavior in passive materials.
Alternatively, if you squeezed it in a vise, it would expand, which would pressurize it even more, which would cause further expansion. Thi... | {
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If fluids exert force in all directions then why do objects float? My question is that when we place a body in water it exerts a buoyant force but it exerts an equal force in downward direction then why its buoyant force is greater and the object floats?
| The force is equal in all directions, but it also depends on the depth. An object of finite size, submerged in a liquid, has its bottom and upper parts at different depths, and the force at the bottom is bigger than that at the top. An object floating on the surface even doesn't have its upper part in the liquid.
| {
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What does squaring a vector mean? So,in the 3rd equation of motion,i.e $v²=u²+2as$,what does the square on the final velocity and initial velocity actually mean?And how does it make an effect on their direction?
suppose upwards vector is positive and downwards vector is negative.And,we are throwing a ball from a height... | $v^2 = v \cdot v$ - where $\cdot$ indicates the dot product.
The result of the dot product is a scalar, not a vector; it doesn't have a direction.
Example for your vector: $\vec{u} = 0 \vec{i} - 4 \vec{j} + 0 \vec{k}$, because $u$ is completely in the vertical direction and there is no component in the horizontal or pe... | {
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In theory, would amplitude keep increasing forever in an oscillating system in resonace? Because there is a limit in experiments modelling resonance, the amplitude of oscillation will eventually reach a limit. However, in theory, if there wouldn't be any limitations in the set up of the system, would the amplitude of o... | Even without damping, you need to have some impulse that transfers the energy. Imagine for example a (poorly built :)) bridge oscillating in a wind; as the motion of the bridge gets faster, the wind imparts less and less energy in the resonance, until it ultimately finds and equilibrium where it's just as likely to add... | {
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What is physically happening in a medium in which an evanescent wave is propagating? Let's consider for instance a metal, for which the dielectric function reads:
$\epsilon = 1 - \frac{\omega_{p}^{2}}{\omega^{2}}$
where $\omega_{p}$ is the plasma frequency. The dispersion relation for an electromagnetic wave is then:
$... | You are correct that the wavenumber becomes purely imaginary. As you approach $\omega = \omega_{pe}$ from the $\omega > \omega_{pe}$ side, the phase speed diverges (i.e., $\tfrac{ \omega }{ k } \rightarrow \infty$) and the group speed goes to zero. Inside the region where $\omega < \omega_{pe}$, the fields do not osc... | {
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Coherent unpolarized laser light I notice that in semiclassical treatments of laser light absorption by particles, they treat the laser beam as a coherent oscillating electric field over the form $E_0\cos(kx-\omega t)$, sometimes with a factor added in to account for the Gaussian spread of wavelengths. However, my unde... | If you take two beam of light and combine them, they can interfere destructively in one place and constructively in another. They will not interfere destructively everywhere. If you follow the details, it always works out that way.
You can see that it has to work out that way by considering conservation of energy. Tota... | {
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Interaction between an electron and an antiproton I have read in many sources that particles and their respective antiparticles annihilate, but a particle and another particle's antiparticle (for example, an electron and an antiproton) do not. However, I could not find what, more exactly, happens when two such particle... | The standard model Lagrangian encapsulates all allowed particle interactions in the weak, strong, and electromagnetic forces. The Lagrangian has individual terms which represent allowed interactions, and these terms can be pictorially drawn as vertices of Feynman diagrams.
For example, the only QED interaction vertex ... | {
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Scattering Amplitude & Unitarity In Srednicki's Quantum Field Theory chapter 11, the probability of a $2 \to n$ scattering process is calculated to be
$$
P = \frac{|\left<f|i\right>|^2}{\left<f|f\right>\left<i|i\right>}
= \frac{(2\pi)^4 \delta\left(k_1+k_2-\sum {k'}_i\right) T}{2E_1 2E_2 V}
|\mathcal{T}|^2\prod_{i'} \w... | In chapter 11 of Srednicki, take note of equation 11.14: $$(2 \pi)^4\delta ^4(0) = \int d^4 x e^{i0\cdot x} = VT,$$ where $V$ is total volume of space (which is cancelled by another $V$ that appears in the denominator to $P$), and $T$ is total elapsed time.
Thus $P \propto T$, so you must divide $P$ by $T$ to get a wel... | {
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Confusion on metrological terminologies in physics I had came across many physicists or engineers use words like Measurable, Detectable, Observable and Quantifiable.
Are those words synonymous in physics in general, in the field of metrology or in engineering?
| No, these are not synonyms, although some if these terms can be - depending on the context.
Quantifiable means that it can be described numerically and compared using greater/lesser/equal operators. E.g., length is quantifiable, but symmetry or presence/absence of a property are not.
Observable often appears in the con... | {
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Dissolution of Carbon Dioxide in water It may sound like an old problem but I don't find a good reference.
Gases are physically soluble in water. Given a water column without any salts solved in it. Air with a given concentration $c_0$ (maybe as mol $\text{CO}_2$ per litre or whatever unit) of carbon dioxide gas is giv... | At equilibrium the entire column will have the same concentration and that concentration will satisfy Henry's law. The only way to have a concentration gradient is if you have a steady removal of CO$_2$ inside the column. The reaction with water you mention cannot be run for ever because they depend on the finite amoun... | {
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Understanding Parallelogram Law of Vector Addition? Recently I've been adding vectors using the Parallelogram Law and the maths is trivial. However, I can't understand the underlying principals. What allows us to move a vector such that the tail meets the head of the other vector? Why can we move the vector to a new st... | According to definition of a vector it is defined as a quantity which have magnitude as well as direction
The reason we can move a vector in this way is because we are not changing its magnitude or direction(essentially the vector remains same). In other words, the vector is simply being translated, or moved to a new p... | {
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Enumerating all the crystallographic directions over a half sphere in a crystal Im taking alook at this paper [1] where there is the following statement:
CO2 was treated as a linear rigid molecule and energies were computed for 61 orientations at each center-of-mass position. These consisted of all the <100>, <110>, <... | It is easier to consider the full sphere first. It is all about finding all the permutations of index and sign allowed by each family of directions.
<100> has 6 possibilities, with a 1 or -1 on any of the 3 axes
<110> has 12 possibilities, with a (1,1) (1,-1) (-1,1) (-1,-1) permuted across 3 axes
<111> has 8 possibilit... | {
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Why do we drop the renormalization term in momentum Klein-Gordon Field Theory? I'm following Peskin & Schroeder's book on QFT. I managed to prove expression (2.33) which gives us the 3-momentum operator for the Klein-Gordon Theory:
$$\mathbf{P}=\int \frac{d^3p}{(2\pi)^3}\mathbf{p}a_\mathbf{p}^\dagger a_\mathbf{p} + ren... | *
*During quantization, classical quantities are typically replaced by normal ordered operators to ensure correct action on the vacuum state, cf. e.g. my Phys.SE answer here.
*Nevertheless, in OP's case of the 3-momentum operator $\hat{\bf P}$ a would-be normal ordering constant
$$ \int_{\mathbb{R}^3}\frac{d^3p}{(2... | {
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Why dont inner molecules pull the stretched soap film inwards? Soap Film Surface Tention
When we stretch the soap film (as shown in linked video), the top and bottom surfaces(shown in blue) get stretched and hence pull back with some force that we call as the surface tention.
But, aren't the inner molecules also stretc... | The surfactant molecules at the surface are not being stretched and this isn't the reason why a surface tension exists. For more on the origin of surface tension see my recent answer to the question Reason for nature of surface tension
Anyhow, the answer to your question is that yes, as the film is stretched the water ... | {
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Can electrostatic precipitation work in reverse? An electrostatic precipitator captures particles by charging them so that they are attracted to a grounded precipitation plate. Does that process work in reverse? I.e., if we apply a voltage to a conductive plate will it repel all particles in contact with it?
(I'm dou... | If the particles are not too heavy or too small then a charged plate can repel them. For example, a paper confetti explosion is a classic Van de Graaff demonstration.
This repulsion can, indeed, be useful.
A recent study - "Electrostatic dust removal using adsorbed moisture–assisted charge induction for sustainable op... | {
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Problem in an ellipse circumscribed on an auxiliary circle I was reading the book "an introduction to the evolution of single and binary stars", by Mattew Benacquista, and I couldn't understand a specific step in topic 2.1 (Time-Depedent Orbits), which says the following:
Consider an ellipse with semimajor axis a that... | This comes from the definition of the ellipse as squished circle along the y-axis.
If the parametric equation of the circle is
$$ \pmatrix{x \\ y} = \pmatrix{ a \cos t \\ a \sin t} $$
which solves the implicit circle function
$$ \left( \frac{x}{a} \right)^2 + \left( \frac{y}{a} \right)^2 = 1$$
then for an ellispe of se... | {
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Does there exist a square root of Euler-Lagrange equations of a field? (Factorization) Does there exist a square root of Euler-Lagrange equations $\partial_{\mu}\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}-\frac{\partial \mathcal{L}}{\partial \phi} = 0$ in the sense that $(x+iy)(x-iy) = x^2+y^2$? By which I... | The Euler Lagrange equations are the equations used to minimise the action. So, they do not exist in versions (i.e. square root of EL eqs etc). What you should be after if you would want (for some reason) a Dirac equation for a scalar field is a Lagrangian whose EL EoM is the Dirac equation.
This is however troublesome... | {
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How to show that the Coriolis effect is irrelevant for the whirl/vortex in the sink/bathtub? There is a common myth that water flowing out from a sink should rotate in direction governed by on which hemisphere we are; this is shown false in many household experiments, but how to show it theoretically?
| The sink demonstration is not an optimal one to show the effect of Coriolis force, mainly because the water in the sink will never be perfectly still enough to start with in order for that force to be the dominant one to determine the direction it will swirl (you could force it to spin either direction.
Better demonstr... | {
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How do physicists use solutions to the Yang-Baxter Equation? As a mathematician working the area of representation of Quantum groups, I am constantly thinking about solutions of the Yang-Baxter equation. In particular, trigonometric solutions.
Often research grants in this area cite this as an "application" of their r... | Ah. Finally a topic I know something about !
There are many places in physics where the YB equation pops up. I can think of two at the moment.
a. Exactly solvable lattice models
b. Quantum Computation (QC)
It is the second application I find most exciting, so I'll focus on it.
The canonical reference (IMHO) on the link... | {
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Is Newton's Law of Gravity consistent with General Relativity? By 'Newton's Law of Gravity', I am referring to
The magnitude of the force of gravity is proportional to the product of the mass of the two objects and inversely proportional to their distance squared.
Does this law of attraction still hold under General ... | Newton's Law of Gravity is consistent with General Relativity at high speed too :)
Lets consider Newton equation of energy conservation for free fall from the infinity with initial speed of object equal to zero:
$\large {mc^2=E-\frac{GMm}{R}}$
or
$\large {mc^2=E-\frac{R_{g*}}{R}\;mc^2}$ where $\large {R_{g*}=GM/c^2}$... | {
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Book about classical mechanics I am looking for a book about "advanced" classical mechanics. By advanced I mean a book considering directly Lagrangian and Hamiltonian formulation, and also providing a firm basis in the geometrical consideration related to these to formalism (like tangent bundle, cotangent bundle, 1-for... | Pure beaten gold, any edition, paperback. It never leaves you. Lagrangian approach.
See the reviews on Amazon
L D Landau (Author), E.M. Lifshitz (Author)
http://www.amazon.com/Mechanics-Third-Course-Theoretical-Physics/dp/0750628960
| {
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What will happen if we add salt to boiling water? I would like to have a good understanding of what is happening when you add salt to boiling water.
My understanding is that the boiling point will be higher, thus lengthening the process (obtaining boiling water), but at the same time, the dissolved salt reduce the pola... | The competition is real, and it's no contest. The reduction in specific heat from the polar effect swamps the miniscule elevation of the boiling point.
When you dissolve salt in water, it makes ions in solution, and the ionic atoms trap a cage of water around them immobile. The net effect is that you reduce the number ... | {
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Intuitively, why is a reversible process one in which the system is always at equilibrium? A process is reversible if and only if it's always at equilibrium during the process. Why?
I have heard several specific example of this, such as adding weight gradually to a piston to compress the air inside reversibly, by why ... | There are two ways to see this. A reversible process is the idealised limit of irreversible processes which go slower and slower. In the limit, a reversible process goes "infinitely slowly" (this is really a phrase used in some thermodynamics texts). I.e., it does not move at all. So the points have to be at equili... | {
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Finding the volume of this irregular shape I have I have an approximately basketball-sized non-hollow piece of aluminum sitting in my house that is of irregular shape. I need to find the volume of it for a very legitimate yet irrelevant reason.
What is the best way I can do this? In fact, what are all the ways I coul... | Measure the mass. If you know it is solid aluminum, then you know its density, and you can easily calculate its volume.
| {
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Is energy really conserved? In high school I was taught energy was conserved. Then I learned that nuclear reactions allow energy to be converted into mass. Then I also heard that apparently energy can spontaneously appear in quantum mechanics. So, are there any other caveats with the conservation of energy?
| Energy is always conserved without any caveat.
With the advent of special relativity, mass and energy are considered equivalent. In other words, they are represented by a vectorial quantity called energy-momentum vector. Before relativity there were separate laws which have been unified.
It is a very fundamental law t... | {
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Planet orbits: what's the difference between gravity and centripetal force? My physics teacher says that centripetal force is caused by gravity. I'm not entirely sure how this works? How can force cause another in space (ie where there's nothing).
My astronomy teacher says that gravity is (note: not like) a 3D blanket... | A centripetal force is a force directed towards the centre. It's just a characterisation of an existing force. "Centripetal" means "towards the centre" in Greek.
So, in the solar system, the sun exerts a gravitational force towards itself, and it is a centripetal force.
Regarding your other question: how does gravity w... | {
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A list of inconveniences between quantum mechanics and (general) relativity? It is well known that quantum mechanics and (general) relativity do not fit well. I am wondering whether it is possible to make a list of contradictions or problems between them?
E.g. relativity theory uses a space-time continuum, while quantu... | Main reason of incompatibility between QM and GR is quantum foam.
Quantum foam are fluctuations on scales below planck length that are so strong that space and time lose its ordinary sence. There the uncertainty principle which causes quantum fluctuations is in straight conflict with smooth spacetime geometry that gene... | {
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What software programs are used to draw physics diagrams, and what are their relative merits? Undoubtedly, people use a variety of programs to draw diagrams for physics, but I am not familiar with many of them. I usually hand-draw things in GIMP which is powerful in some regards, but it is time consuming to do things ... | To start with, for scientific drawing usually vector graphics is more suitable - scalable, convenient to modify and produce less bulky files.
For simple general-purpose graphics I use OpenOffice.org Draw (I prefer it to Incscape).
For abstract diagrams there is yEd - Graph Editor.
Both are free, for Win/Linux/MacOSX, e... | {
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Why do we automatically assume that the velocity vector $\vec{v}$ and location vector $\vec{r}$ are independent? I'm not sure if it's relevant, but I'm talking about a situation where a particle is moving in an electro-magnetic field.
As I understand, if we see the term $\nabla \cdot \vec{v}$ or $\nabla \times \vec{v}$... | For a single particle motion $div(v) = [div(dr/dt)] = d/dt [div(r)]$. $div(r) = 3 = const.$ But the time-derivative of a constant is zero, so $div(v) = 0$. It is not an automatic assumption, it is a calculation result ;-).
EDIT: somebody voted down. OK, v = v(t) so any $\partial v_i/\partial x_k = 0$ by definition of v... | {
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Distance travelled in free-fall When an object is in free fall, we have:
$$a(t) = g - \frac{c}{m}v(t)^2$$
where $g$ is acceleration due to gravity, $m$ is the mass of the object, and $c$ is the coefficient of air resistance.
How does one get the distance traveled after t seconds? I tried integrating it, giving
$$v(t) =... | The equation you're looking for is
$$m\frac{dv}{dt}=\frac{1}{2}\rho C_{D} A v^{2}-mg$$
where $C_{D}$ is the drag coefficient and $A$ is the cross-sectional area of the object.
For the vertical position as a function of time, you can see the solution in Free Fall.
By the way, your calculation is not quite right. You ha... | {
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4th kinematic equations for constant acceleration Ive been given the 4 kinematic equations for constant acceleration. The fourth being:
$$s=ut+\frac{1}{2}at^2.$$
If rearranged it forms the quadratic equation
$$at^2+2ut-2s=0.$$
But that means that $t$ has 2 values.
*
*Will one of them always be negative?
So only on... | When you throw a ball into the air, there are two times it will have a certain height $s$. Once going up and once coming down. That is what the two times represent.
Both are valid.
| {
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How are neutrons produced from cosmic ray particles? What are the details of how neutrons are produced as a result of cosmic ray particles hitting our planet's atmosphere?
For instance, what is the pathway that creates the highest number of neutrons from cosmic ray particles?
The article "Single event upset" states:
A... | Between what I've learned about cosmic rays and what I can find online (example: http://www.fisica.unlp.edu.ar/~veiga/experiments.html), it seems that the primary source of neutrons in cosmic ray showers is the disintegration of the atomic nuclei that are struck by the cosmic ray or its decay products.
As you may know,... | {
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Why does kinetic energy increase quadratically, not linearly, with speed? As Wikipedia says:
[...] the kinetic energy of a non-rotating object of mass $m$ traveling at a speed $v$ is $\frac{1}{2}mv^2$.
Why does this not increase linearly with speed? Why does it take so much more energy to go from $1\ \mathrm{m/s}$ to... | As Piotr suggested, accepting the definition of work $W=\mathbf{F}\cdot d\mathbf{x}$, it follows that the kinetic energy increases quadratically. Why? Because the force and the infinitesimal interval depend linearly on the velocity. Therefore, it is natural to think that if you multiply both quantities, you need to end... | {
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Material resistency to lasers beam Keeping the average power constant, why some materials are more eager to be damaged by pulsed laser with respect of C.W. lasers, or viceversa?
When i talk about pulsed lasers i think for examples of duty cycles in the order of $10^5$.
For example optical elements (such as a vortex pha... | Typically, a laser will damage an optical surface in one of two ways. The first is just what you would expect: the laser heats the material up until something bad happens. The second is also pretty simple, but less common because (AFAIK) it is really only a problem with very short pulses (on the order of femtoseconds).... | {
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Evolution in the interpretation of the Dirac equation As I understand, Dirac equation was first interpreted as a wave equation following the ideas of non relativistic quantum mechanics, but this lead to different problems.
The equation was then reinterpreted as a field equation and it is now a crucial part of quantum f... | I recommend J.D. Bjorken, S.D. Drell, Relativistic Quantum Mechanics.
I think this book can provide smooth introduction into Dirac equation - the book is almost dedicated to it. At the same time I discourage starting with Quantum Field Theory before learning Dirac equation - the conceptual step may be way to difficult.... | {
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What if the size of the Universe doubled? My question has a silly formulation, but I want to know if there is some sensible physical question buried in it:
*
*Suppose an exact copy of our Universe is made, but where spatial distances and sizes are twice as large relative to ours. Would this universe evolve and funct... | From what I know of cosmology, we are only able to see a small (and shrinking) portion of a much bigger universe. The only real measures of size (I think) are age (from the hubble constant) and density. If we wait for the age to roughly double (the dark energy expansion means we may get there a bit sooner), we will be ... | {
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Suggested reading for renormalization (not only in QFT) What papers/books/reviews can you suggest to learn what Renormalization "really" is?
Standard QFT textbooks are usually computation-heavy and provide little physical insight in this regard - after my QFT course, I was left with the impression that Renormalization ... | Renormalization is absolutely not just a technical trick, it's a key part of understanding effective field theory and why we can compute anything without knowing the final microscopic theory of all physics. One good online source that explains a nice physical example is Joe Polchinski's "Effective field theory and the ... | {
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Is Dr Quantum's Double Slit Experiment video scientifically accurate? I'm fascinated by the fundamental questions raised by the Double Slit Experiment at the quantum level. I found this "Dr Quantum" video clip which seems like a great explanation. But is it scientifically accurate?
| A bad thing about the video is how they explained the part where you try to observe which slit the electron goes through. They made it sound more mysterious than it really is.
What we have to ask ourselves is: what does it mean to observe an electron? What does it mean to observe anything? If we want to look at somethi... | {
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Books that every physicist should read Inspired by How should a physics student study mathematics? and in the same vein as Best books for mathematical background?, although in a more general fashion, I'd like to know if anyone is interested in doing a list of the books 'par excellence' for a physicist.
In spite of the ... | There is a lot of good books in CM, QM, EM... but what every physicist should read, undoubtly, are The Feynman Lectures on Physics.
| {
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Relativistic Cellular Automata Cellular automata provide interesting models of physics: Google Scholar gives more than 25,000 results when searching for "cellular automata" physics.
Google Scholar still gives more than 2.000 results when searching for "quantum cellular automata".
But it gives only 1 (one!) result when ... | Lattice Boltzmann take on 1+1 dimensional quantum field theory:
*
*S. Succi, Lattice Boltzmann method for quantum field theory, 2007 J. Phys. A: Math. Theor. 40 F559
| {
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Lightning strikes the Ocean I'm swimming in - what happens? I'm swimming in the ocean and there's a thunderstorm. Lightning bolts hit ships around me. Should I get out of the water?
| In fresh water what makes lightening so dangerous to a swimmer is that most of the current travels on the surface of the water, so rather then getting a $1/r^2$ falloff in current density, you see a $1/r$ falloff. Obviously eventually it will be conducted down into the mass of the water, but this takes a many meters. I... | {
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How does gravity escape a black hole? My understanding is that light can not escape from within a black hole (within the event horizon). I've also heard that information cannot propagate faster than the speed of light. It would seem to me that the gravitational attraction caused by a black hole carries information ab... | The relativistic formula for the radius of a black-hole is; r=2GM/c^2. The Newtonian escape velocity is; r=2GM/v^2, where v is the velocity of the projectile. We see that the two are the same after replacing v by c. The fact that radiation is massive or massless doesn't make a difference as it cancels from the final eq... | {
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Why isn't dark matter just ordinary matter? There's more gravitational force in our galaxy (and others) than can be explained by counting stars made of ordinary matter. So why not lots of dark planetary systems (i.e., without stars) made of ordinary matter? Why must we assume some undiscovered and unexplained form of m... | You can estimate total mass using gravitational lensing and compare it to the estimated mass of a galaxy by adding up the masses of all the visible stars and making generous estimates of other kinds of ordinary matter that could be there. There is still a big discrepancy.
If the extra mass were ordinary matter, it woul... | {
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Common false beliefs in Physics Well, in Mathematics there are somethings, which appear true but they aren't true. Naive students often get fooled by these results.
Let me consider a very simple example. As a child one learns this formula $$(a+b)^{2} =a^{2}+ 2 \cdot a \cdot b + b^{2}$$ But as one mature's he applies t... | That the PI Team is the best group to design and implement their own data systems, rather than bring in people experienced with IT security, data modeling for reuse by other groups, and other data informatics issues.
Related misconception: that their data is so different from other data that a data system must be desig... | {
"language": "en",
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What is the evidence for Inflation of the early universe? The theory of Inflation explains the apparent consistency of the universe by proposing that the early universe grew exponentially for a 1E-36 seconds. Isn't a simpler explanation that the universe is just older and so the homogeneousness comes from a slower mor... | The main (and original) reason for the proposal of inflationary theory was the horizon problem. That is, the fact the the universe is so incredibly homogeneous and isotropic despite the fact that some parts of the universe are apparently too far away to have exchanged energy. Inflation in the early universe is a powerf... | {
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How do electrons choose their path? Wire A
-----R-----
/ \
------I--->-[P]-- ---------I-->
\ /
-----R-----
Wire B
Wire A and B both have a resistance of $R$ and a curren... | Quantum effects, as others have pointed out are negligible in most wires. If you want to derive from a particle like description of the wires, you'd be best off thinking in terms of gas/fluid dynamics (which is believed to in the continuous limit turn into the transport diffusion equations). You can simulate this on a ... | {
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What is the difference between "kinematics" and "dynamics"? I have noticed that authors in the literature sometimes divide characteristics of some phenomenon into "kinematics" and "dynamics".
I first encountered this in Jackson's E&M book, where, in section 7.3 of the third edition, he writes, on the reflection and r... | I'd say that kinematics has to do with the space of all possible configurations of a system at one time, for example what restrictions are placed on those by conservation laws. Dynamics has to do with how configurations change as a function of time. As always, the way the term is actually used may depend on the person.... | {
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Are gauge choices in electrodynamics really always possible? If $B$ is magnetic field and $E$ electric Field, then
$$B=\nabla\times A,$$
$$E= -\nabla V+\frac{\partial A}{\partial t}.$$
There is Gauge invariance for the transformation
$$A'\rightarrow A+{\nabla L}$$
$$V'\rightarrow V-\frac{dL}{dt}.$$
Now, we can write:
... | The correct Gauge transformation formula should be
$$\begin{aligned} \mathbf A &\mapsto \mathbf A + \nabla \lambda \\
\mathbf V &\mapsto V - \frac{\partial\lambda}{\partial t}, \end{aligned} $$
not something with "gradL/dt". The Coulomb gauge requires $\nabla\cdot\mathbf A=0$, not "rotA = 0". The Lorenz gauge requires ... | {
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Doppler's effect use While i was in high-school i learn't the Doppler's Effect which if i remember correctly is:
*
*The Apparent change in the frequency of sound caused due the relative motion between the sound and the observer.
This phenomenon seems obvious, but what i would like to know is, what use does Doppler... | I haven't used it much, but you can use it in games development to implement 3D sound.
| {
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"timestamp": "2023-03-29T00:00:00",
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Utility of displacements potentials in geophysics In the elasticity theory, you can derive a wave equation from the fundamental equation of motion for an elastic linear homogeneous isotropic medium:
$\rho \partial^2_t \overline{u} = \mu \nabla^2 \overline{u} + (\mu+\lambda) \nabla(\nabla \cdot \overline{u})$
But in th... | I believe potentials in Maxwell's equations were introduced originally to make solving equations simpler -- it wasn't until a bit later that the Lorentz symmetry was noticed. Similar potentials are introduced in 2D hydrodynamics as well, see e.g. stream function. So I would say that the reason for introducing these po... | {
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Best example of energy-entropy competition? What are the best examples in practical life
of an energy-entropy competition which favors entropy over energy?
My initial thought is a clogged drain -- too unlikely for the
hair/spaghetti to align itself along the pipe -- but this is probably
far from an optimal example. Cu... | One example I know is so-called Brazil nut effect. When you place balls in a container of two different sizes and shake it, the larger ones will go up (even if they are denser than the smaller ones). So the final energy of system after introducing noise is clearly greater than the initial. I believe that the phenomena ... | {
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What is the definition of momentum when a mass distribution $\rho(r,t)$ is given? This question is Edited after recieving comments.
What is the definition of momentum when a mass distribution $\rho(r,t)$ is given?
Assuming a particle as a point mass we know the definition of momentum as $p = mv$.
I need a definition w... | Motion of a non-point object can be described as a combination of translation (which only depends on the total mass of the object) and rotation (which depends on the mass distribution). Rotational momentum is defined as $\mathbf{L}= I \boldsymbol{\omega}$ where $I$ is a moment of inertia $I = \int_V \rho(\mathbf{r})\,d... | {
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Is it possible to obtain gold through nuclear decay? Is there a series of transmutations through nuclear decay that will result in the stable gold isotope ${}^{197}\mathrm{Au}$ ? How long will the process take?
| The following wikipedia link has a table of all the known isotopes of all the known elements. Gold has Z=79. According to the table, there's only one stable isotope of gold, $\;^{197}Au$ :
http://en.wikipedia.org/wiki/Table_of_nuclides_%28complete%29
From this you can figure out which isotopes decay into gold by consid... | {
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Which experiments prove atomic theory? Which experiments prove atomic theory?
Sub-atomic theories:
*
*atoms have: nuclei; electrons; protons; and neutrons.
*That the number of electrons atoms have determines their relationship with other atoms.
*That the atom is the smallest elemental unit of matter - that we ca... | I once heard Uhlenbeck give a lecture on this to high school students over the Christmas break at the Rockefeller Univ. years ago. He recounted a published argument he attributed to Einstein around 1905 (I think), which was that atoms were real if you could count the number of them/mole (Avogadro's number) many diffe... | {
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Applications of Algebraic Topology to physics I have always wondered about applications of Algebraic Topology to Physics, seeing as am I studying algebraic topology and physics is cool and pretty. My initial thoughts would be that since most invariants and constructions in algebraic topology can not tell the difference... | Twistor theory uses sheaf cohomology, see e.g. Gentle introduction to twistors.
And twistor theory itself has applications to perturbative quantum field theory (MHV amplitudes).
| {
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What's the difference between running up a hill and running up an inclined treadmill? Clearly there will be differences like air resistance; I'm not interested in that. It seems like you're working against gravity when you're actually running in a way that you're not if you're on a treadmill, but on the other hand it s... | Let's estimate some of the contributions discussed in dmckee's answer
Gravity
We can compute the power spent gaining altitude.
$$ W_\text{grav} = mg \dot h = m g v \sin \theta \sim m g v \theta $$
for small angles, with some typical numbers
$$ W_\text{grav} = ( 180 \text{ lbs} ) ( 9.8 \text{ m/s}^2 ) ( 1 \text{ mile} /... | {
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Is it possible to accelerate air to supersonic speeds? What would it look like? The speed of sound is the rate that disturbances in air propagate through it.
Is it possible to have a wind that itself is moving at supersonic speeds relative to stationary winds around it?
Or perhaps a fluid flowing through a pipe at a sp... | Read something about Laval nozzles. In a narrow part of the nozzle the air attains the sound velocity and in a divergent part (diffuser) it is accelerated even more.
Deceleration of such supersonic flow is only possible in a shock wave (suddenly), not gradually! There are plenty of pictures of supersonic flows from no... | {
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"answer_id": 1
} |
What is the range of validity of Fermi's Golden Rule? It is well known that to calculate the probability of transition in the scattering processes, as a first approximation, we use the Fermi golden rule. This rule is obtained considering the initial system in an eigenstate of the unperturbed Hamiltonian $H_0$ and consi... | Here:
...the assumption that the time of the measurement is much larger than the time needed for the transition...
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/1766",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
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