Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Where did the work done by smaller force go? Suppose I have a spring of spring constant 150 N/m.One person pulls with it a force of 15 N. The extension produced is 0.1 m. Now another person comes and pulls with a force of 30 N (The first person is still there). The final extension is 0.3 m.The initial and final potenti... | You are trying to compare the work done by constant forces to the change in potential energy of the spring. However, if you take a spring at rest at its current equilibrium and then apply a purely constant force to it, then the spring will not come to rest at a new equilibrium position; it will start oscillating about ... | {
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Effects of Couloumb’s law on electron beam In an electron beam there are about $10^{15}$ electrons. If they are all electrons, then they all have the same charge, so why don't they repel each other? Furthermore, the force would extend to infinity because it is inversely proportional to the distance.
Where is that force... | Firstly, the exclusion principle does not say that electrons repel each other. It’s states that fermions can not occupy the same quantum state. The state depends on more than just charge and position. It also depends on spin. And, although the consequences of the exclusion principle mean that electrons don’t get too cl... | {
"language": "en",
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Which force makes the contact point of an object in rolling motion rise? When an object is rolling without slipping, the point at the bottom has no tangential velocity. When the point at the bottom rotates and rises, what force lets the point rise with tangential velocity from having no tangential velocity from before?... | You have to be careful about two different ways of thinking about the object.
In problems like this, the object is treated as a single rigid thing. You ask what are the force on the object to learn how it as a whole accelerates. The forces you consider are typically gravity, the force of the floor or inclined plane ho... | {
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What is the luminosity of the Milky Way? This question may appear to be the same as What is the luminosity of the Milky Way galaxy?, but that question references a value of $5\times 10^{36}\space W$, which comes from a Wiki page which references an article from 1999. While Rob Jeffries provides some excellent insight ... | As far as I know, the best current estimate is that the Milky Way galaxy has an absolute magnitude of
$$^0M_r - 5 \log h = -21.00^{+0.38}_{-0.37}$$
which is equivalent to a luminosity of $5.4 \times 10^{36}$ to $1.1 \times 10^{37}$ W. This makes the Milky Way one of the brightest known spiral galaxies.
| {
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In a vacuum, can you see light which is not travelling towards you?
In air, when there is light propagating in a direction, we can still see it even when it is not primarily travelling in our direction, because a small part of the light hits the air molecules, and changes its direction; it travels towards us.
Does thi... | As the other answers state, the answer is no, if there is nothing for light to scatter off of, to change its direction towards you, you will not see anything.
I would like to add, that it might be interesting to consider two things:
There is no perfect vacuum, and the real vacuum in our universe does have certain parti... | {
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Gauge transformation and phase matching in cylinder coordinate for the Aharonov-Bohm Effect I'm trying to get through Sakurai The Aharonov-Bohm Effect where on page 141.
According t0 Eq 2.7.53, if the original equation was transformed by a gauge $\tilde A=A+\nabla \Lambda$, then the solution for the new system was just... | Look at problem 28.c
If you require the ground state energy to be unchanged by the presence of $B$, then you require the boundary condition to hold.
So:
$$\frac{e}{\hbar c}\frac{B \rho_a^2}{2} = m \in \mathbb{Z} \quad \implies\quad\pi \rho_a^2B=\frac{2\pi m\hbar c}{e}, $$
which is known as flux quantisation.
Experiment... | {
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Can we apply simple mechanics laws on sound waves I made up a question to make you understand my doubt , see
Assume that a car is travelling with speed v to a wall . When car is at distance x from wall it emits a sound signal . Find time after the signal reaches the car again ? ( Speed of sound $ = v_s$ )
Now if we th... |
Now if we think that sound wave is like an object and use relative motion than sound will approach wall with speed $v + v_{s}$...
No, I think you are misunderstanding something here. A linear sound wave will always propagate at the speed of sound once emitted in a homogeneous, uniform medium. If your expression wer... | {
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Dirac action momenta conjugate to conjugate field Consider the Dirac action $S=\int d^4x\bar{\psi}(x)(i\not\partial-m)\psi(x)$. Since there are no time derivative of $\bar{\psi}$, we get the constraint that its canonical momenta vanishes. This constraint is of course first class. Does this mean that the Dirac equation ... | The are 2 primary constraints rather than 1. (Recall that there is also 1 primary constraint for the complex conjugate field). The 2 primary constraints do not (super)Poisson-commute, so they are second-class rather than first-class constraints. And therefore no gauge symmetry. See e.g. my related Phys.SE answer here.... | {
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Another question of index notation in tensor calculus I have this equation
$$\nabla_{a}(g_{bc}\lambda^{c})=(\nabla_{a}g_{bc})\lambda^{c}+g_{bc}\nabla_{a}\lambda^{c}$$
And making some calculations
$$ \lambda^{c} (\nabla_{a}g_{bc})= \nabla_{a}(g_{bc}\lambda^{c})- g_{bc}\nabla_{a}\lambda^{c} $$
$$ \lambda^... |
I want to get an expression for $\nabla_{a}g_{bc}$
If you are using the Levi-Civita connection, then the covariant derivative of the metric is zero by the definition of that connection.
See this PSE question.
If you don’t know which connection you’re supposed to be using, you’re almost certainly supposed to be using ... | {
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Total number of microstates in a system If we consider a system made of two isolated parts, say A and B where let's say A has 100 particles and B has 100 particles. If the coordinates of ($q$,$p$) specify the microstate of a particle, then while calculating the total number of microstates, why we multiply the microstat... | For each one of the $\Omega_A$ possible arrangements of the first subsystem, there are $\Omega_B$ passible arrangements of the second. Similarly, if I roll a die with my left hand and a die with my right, the number of possible combinations is not $6+6=12$, but rather $6\cdot 6=36$.
| {
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Is the potential energy stored in a spring proportional to the displacement or the square of it? Suppose a mass of $M$ kg is hanging from a spring in earth. The mass will stretch the spring about $x$ m. So the change in the gravitational potential energy is $mgx$ J (supposing $x$ to be very small compared to the radius... | If there is just the (ideal) spring and a mass, i.e., if there is no dissipation, the total energy $E$ of the system is constant and is the sum of three terms:
$$E = mgh + \frac{1}{2}kz^2 + \frac{1}{2}m\dot{z}^2 = U_g + U_s + T$$
where $h$ is the height of the mass from the ground (zero reference of gravitational poten... | {
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How does cutting a spring increase spring constant? I know that on cutting a spring into n equal pieces, spring constant becomes n times.
But I have no idea why this happens.
Please clarify the reasons
| In general, the spring constant or stiffness of a coiled spring is given as
$$k=\frac{\pi Gd^4}{64R^3n}$$
Where, $G$ is modulus of rigidity of spring material
$d$ is the diameter of spring wire
$R$ is the mean radius of coil
$n$ is the effective number of coils in spring which is directly proportional to the length o... | {
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Would a perfectly collimated laser beam have a flat intensity distribution? I'm trying to simulate a collimated laser beam in some FORTRAN and am wondering about the intensity distribution of a perfectly collimated beam: would it be a flat distribution (equal intensity across the beam) or it would be more of a gaussian... | Collimation means that all pencil ("bundle") of light rays emitted from one specific point in the object plane is "translated" by the collimator to a pencil of rays that have a specific angle.
In this case, the object is the output surface of the laser cavity or the output of an optical fiber.
Imagine you have a point ... | {
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Does the centre of charge behave as a point charge? We can find the centre of charge of a system of charges in much the same way we find the centre of mass of a system of masses
Suppose we have two charges of the same polarity and equal in magnitude that are placed at -x and +x along the x axis. The centre of charge wo... | For an external point in the space, an electric charge uniformly distributed over the surface of an object behaves like a point charge of same magnitude located at the geometric center of that object.
Example: A uniformly charged (metallic or non-metallic) sphere behaves like a point charge for all the external point ... | {
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Mathematically prove that a round wheel roll faster than a square wheel Let's say I have these equal size objects (for now thinking in 2D) on a flat surface.
At the center of those objects I add equal positive angular torque (just enough to make the square tire to move forward).
Of course the round tire will move fast... |
Let's say I have these equal size objects ...
First counterquestion: What does "equal size" mean?
*
*The diameter of the circle is the same as the edge length of the square
*The square and the circle have the same area
Mathematically prove that round objects roll faster
Even in the first 90 degrees where we d... | {
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What is the range of Pauli's exclusion principle？ In many introductions to the pauli's exclusion principle, it only said that two identical fermions cannot be in the same quantum state, but it seems that there is no explanation of the range of those two fermions. What is the scope of application of the principle of exc... | All electrons (and all elementary particles) in the universe are supposed to have exactly identical properties according to the standard model. This means that for electrons, the Pauli exclusion principle reads "No 2 electrons in the universe can occupy the same state".
But due to the phrasing of your question, I think... | {
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Physical meaning of Cavity Finesse I was wondering what does the term cavity finesse mean, physically? I understand that it is defined by—
$$F = \frac{ \text{Free Spectral Range (FSR)}}{\text{Linewidth}},$$
where FSR is the frequency separation of the cavity modes given by the expression $\frac{\pi c}{nd}$, $d$ being t... |
that finesse is the measure of the cavity losses and is independent of resonator separation
I guess I would agree with that definition, but would add the word relative before "measure".
The FSR is dictated by the resonator geometry. A shorter resonator will have a larger FSR, i.e. the peaks are spaced farther away.
T... | {
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Is 1 Joule the work done to lift ~100g through a distance of 1m? I am seeing many videos saying that 1 Joule is the work done to lift ~100g through a distance of 1m (like this one https://youtu.be/BYpZSdSEk4A?t=348).
The idea is that 100g has a gravitational force downward of 1 Newton. So lifting it means applying 1 Ne... | Yes, to start lifting the object you will need to apply a force larger than its weight so that it will have an upwards acceleration. This force does not have to be much large than the weight, it could be larger by an arbitrarily small amount, it just depends on how much time you want to spend lifting. The lifting for... | {
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Must conclusions from relativistic physics hold in non-relativistic physics? Example:
When one studies the spin statistics theorem, one of the phrases that's been repeated a lot was that "the spin statistics theorem was derived from relativistic physics... there's no way to prove it in non-relativistic physics."
Howev... | A simple counterexample: There is no frame in GR where gravitation travels with infinite velocity.
| {
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Why is the Force of Gravitational Attraction between two “Extended” bodies proportional to the product of their masses? Newton’s Law of gravitation states that force of attraction between two point masses is proportional to the product of the masses and inversely proportional to the square of the distance between them.... | The statement
...the force between Any two rigid masses is only proportional to the
product of their masses
is not true in general, or at least it is misleading. The shapes of the mass distributions and their relative positions matter when computing the gravitational force.
It is true that once you hold constant th... | {
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Confusion about an example of inertia related to bus When the bus is stationary the passengers are also stationary. When the bus starts moving the part of the body (lower part) in contact with bus starts moving, but due to inertia the upper part remains stationary and thus he gets reclined back. If he is standing he wi... |
I cannot understand that how lower part starts moving while upper part
remains stationary?
First of all, the person is not rigid body. It can flex and bend in response to forces acting and not acting on different parts of the body.
Given that, the static friction force between the standing persons feet and floor of... | {
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How do "see through mirrors" work? (I called them see through mirrors because I don't know what their actual name is)
I have seen a lot of pictures of glass-like materials which reflect light from on side, and from the other side, they transmit light.
What i want to say is -
Let side A and side B be the two sides. Si... | One way mirrors are just semi-transparent mirrors with a little trick.
The laws of thermodynamics prevents a "real" one-way mirrors from existing; if they did exist, then we can put two rooms at the same temperature on the two sides of it. Then the radiation from one side will pass through but the radiation from the ot... | {
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Why is electric flux through a cube the same as electric flux through a spherical shell? If a point charge $q$ is placed inside a cube (at the center), the electric flux comes out to be $q/\varepsilon_0$, which is same as that if the charge $q$ was placed at the center of a spherical shell.
The area vector for each inf... |
Why does electric flux through a cube is same as that of electric flux through a spherical shell?
This is not only true for a cube or a sphere. The flux passing through any closed surface enclosing a net charge $q$ is $q/\varepsilon_0$. This is based on Gauss's law for electric charges.
When the field lines emerge f... | {
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How to get the formula of the energy of EM waves? I am trying to get the formula for energy of EM waves:
$$W = \frac{E^2 + B^2}{2}$$ calculating the work done on a test charge by the force: $$\mathbf F = q(\mathbf E + v \times \mathbf B)$$ $\mathbf E$ and $\mathbf B$ are vectors of the type $\mathbf F(u)$, $u = (\math... | The usual approach is to calculate the power due to Joule's heat, given by $\mathbf{j}\cdot\mathbf{E}$. Starting with the Maxwell equations
$$\nabla \times\mathbf{B}=\frac{4\pi}{c}\mathbf{j} + \frac{1}{c}\frac{\partial\mathbf{E}}{\partial t},\\
\nabla \times\mathbf{E}=- \frac{1}{c}\frac{\partial\mathbf{B}}{\partial t}.... | {
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Can you elaborate the concept of symmetry and how do you conclude that the field is $0$ along $x$-$z$ plane? This is a fairly popular problem.
An infinite slab, of thickness 2d, carries a volume charge density $\rho$.Find the electric field, as a function of $y$, where $y=0$ at the center. Plot $E$ versus $y$ calling ... | I thought it wasn't defined in $ y = 0 $, but here is my argument anyways.
In general, bilateral symetry of a charge configuration in a plane $ x = k $, implies that the field has no $\hat{x}$ component on $ x = k $.
In the charged plane example, you have bilateral symetry for all $ z = k_1 $ and $ x = k_2 $ (that's w... | {
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Is a quantum eraser instantaneous? I'm going through the basic and popular optical quantum-eraser experiment with laser and polarizers and I have a question.
After applying the diagonally aligned polarizer at a 45 degree angle, the B1 section of the beam carries an interference that can be observed by the fringes at t... | This is a pretty bad example of a quantum eraser experiment, because it can be explained entirely using classical electromagnetism. Specifically, the two concepts involved are:
*
*Malus's Law, which describes the output intensity of light as it passes through a linear polarizer, and
*The Fresnel-Arago laws, which,... | {
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Is the average force calculated from $F(x)$ the same as that calculated from $F(t)$? Say a force is doing work on an object in one dimension. I could calculate the average force over the distance with
$$\frac{1}{\Delta{x}}\int_{x_1}^{x_2} F(x) \text dx$$
If I also formulated force as a function of $t$, I could calculat... | No. They aren’t the same. To see why consider the displacement (1D) being parametrised by time in a way to give $x(t)=\alpha t^2$ where alpha is a constant to fix the dimensions. From Newton’s second law this gives us $F(x)=2\alpha m$, then,
$$\frac{1}{\Delta x}\int F(x)dx= \frac{1}{\Delta x}\int_{x(t_1)}^{x(t_2)} 2\al... | {
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Water Evaporation on Earthen Pot In earthen pot water molecules lose heat energy, so where is the temperature drop more - on the pot's surface or inside water?
| For water to evaporate faster in a pot, heat must flow into the system (of water) from the pot more than the flow of heat from it to the pot. And we know that heat flows from hotter object to colder. Thus the water must remain at a lower temperature.
However, soon an equilibrium will be established between the water a... | {
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Acceleration in photoelectric effect In all the books I read, only the initial speed with which an electron is removed from a plate is mentioned, what happens to the acceleration?
Consider a plate (charge neutral and not connected to anything) and light of appropriate wavelength hits the plate
Suppose that an electron ... | Suppose that you have a neutral plate whose potential relative to the "ground" is zero.
As photo-electrons are emitted from the plate, the potential of the plate increases, and so subsequent photo-electrons are attracted to the plate but if they have sufficient kinetic energy they will escape from the plate.
Howeve... | {
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What is meant by "maximum amount of information is available" in quantum measurements? I’m reading Isham’s Lectures on Quantum Theory, and in Chapter 5, General Formalism of Quantum Theory, Section 5.1.2, he states the following rule.
Rule 1. The predictions of results of measurements made on an otherwise isolated sys... | Probably the author means that a state in which you know everything you can know about that system, is represented by such a vector (or rather a ray, as Alfred Centauri remarked).
When there is uncertainty in the state, which could be considered classical uncertainty (where quantum uncertainty would be the uncertainty ... | {
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Three operators commuting with each other It is well known that if two operators commute, the it is possible to find common eigenfucntions for them. What if we have 3 operators that commute with each other? Will we find common eigenfunctions for all of them at the same time?
| Yes. For example, when calculating angular momentum of a hydrogen atom (ignoring fine structure or hyperfine structure), the eigenbases (which are described by spherical harmonics) are simultaneously eigenbases of the Hamiltonian operator $H$, the total angular momentum operator $L^2$, and the z-component angular momen... | {
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What happens to an inductor if the stored energy does not find a path to discharge? Suppose an inductor is connected to a source and then the source is disconnected. The inductor will have energy stored in the form of magnetic field. But there is no way/path to ground to discharge this energy? What will happen to the s... | The inductor becomes an active inductor. The energy is still stored in it, and the total flux it produces remains the same.
If you connect it to another circuit, (Say, with just a resistor), it will momentarily act as a source of current i.e. the first current that flows through the circuit will be the same as the one... | {
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Since computer screens can't display many colors, where can I go to see the full color gamut? I found out recently that computer screens are only able to display a subset of all the colors visible to the human eye. Naturally one of my first questions was what do the other colors look like, but this is one of the few qu... | You can't go anywhere to see the "full color gamut". There are three types of cone cells on your retina that only see the colors red, green, and blue, which are the same colors that are imprinted into the tiny phosphors on your color TV screen. Thus, your eye can only see these three colors.
In everyday experience, a... | {
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Force on dielectric slab My Doubt is regarding the procedure of calculating force on dielectric while it is being inserted between the capacitor plates (with or without battery).
This is the theory behind it.
Consider a capacitor which is charged and disconnected from the battery.
Now the energy stored in the capacito... | To quote Griffith's E&M Text (3rd edition pg. 196)
Notice that we were able to determine the force without knowing anything about the fringing fields that are ultimately responsible for it! Of course, it's built into the whole structure of electrostatics that $\nabla\times\mathbf E=0$, and hence that the fringing fiel... | {
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Formula for period of pendulum using energy conservation I'm trying the derive the period of a simple pendulum using energy conservation and without calculus.
I'm doing something wrong which I can't figure out.
I see a lot of other derivations online using calculus which I want to avoid for now.
The pendulum has a leng... | It can be shown geometrically that the pendulum angle is θ = sqrt(2h/L) where h is the height of the mass. Then the energy equation
(1/2)L^2 ( θ' )^2 + g L ( θ^2 )/2 = constant. (θ') is first derivative of θ with respect to t.
We now take the time derivative of the energy expression and it quickly simplifies to
L... | {
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Limitations/breakdown of Mermin-Wagner Theorem Mermin-Wagner theorem says that continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions $d ≤ 2$. (this is directly copied from wiki).
I'm just wondering that, if we could add some intera... | $\newcommand{\pd}{\partial}\newcommand{\d}{\mathrm{d}}$I bring back this old question to supplement the other answer with what happens at $d\geqslant 3$. A version of Mermin–Wagner valid for higher dimensions can be formulated using higher-form symmetries.
Let me first remind you what higher-form symmetries are. Since ... | {
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Potential function for conservative internal forces In Goldstein's Classical Mechanics book, he considered a system of particles and looked at the conservative internal force between particle $i$ and particle $j$ that satisfy the strong law of action and reaction.
He wrote the potential function for this internal forc... | Mathematically is just the chain rule?
Since
\begin{align}
\dfrac{\partial}{\partial \vec{r}_i}V_{ij}(\vert \vec{r}_i-\vec{r}_j\vert)&=\dfrac{\partial V_{ij}}{\partial (\vec{r}_i-\vec{r}_j)}\dfrac{\partial (\vec{r}_i-\vec{r}_j)}{\partial \vec{r}_i}\\
&=\dfrac{\partial V_{ij}}{\partial (\vec{r}_i-\vec{r}_j)}(1-\delta_... | {
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Quantum vs Classical estimation using Fisher Information Is it possible to have a different estimated value of a parameter by Classical Fisher Information (CFI) compared to Quantum Fisher Information (QFI)?
Say, for example, I am plotting, $\mathcal{F_{\nu}} $, Fisher Information for parameter $\nu$ with respect to $\... | The key difference between the QFI and CFI is that CFI depends on the measurement you use.
Is it possible to have a different estimated value of a parameter
Neither of these are estimators, they give you bounds on your estimate so I'm not 100% what you mean here. You would use say, maximum likelihood estimation if yo... | {
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Proof for elliptical orbits Its mentioned in several books that a satellite launched with a velocity less than the escape velocity and other than the critical velocity will follow an elliptical orbit. However I can't find a derivation of its equation of trajectory.
| This problem is most easily done in polar coordinates. In polar coordinates, the Newtonian gravity is given by $$\mathbf{g} = -\frac{GM}{r^2} \mathbf{\hat{r}}$$
Applying Newton's Second Law in polar coordinates gives two differential equations: one radial and one angular: $$\ddot{r} - r \dot{\theta}^2 = -\frac{GM}{r^2}... | {
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Jacobian rules with canonical transformations
If we consider a canonical transformation from $(q,p)$ to $(Q,P)$, it is stated in several sources that by Jacobian rules,
$$ \frac{\partial(Q,P)}{\partial(q,p)} = \frac{\partial(Q,P)/\partial(q,P)}{\partial(q,p)/\partial(q,P)}. \tag{1}
$$
By taking books such as Riley'... | The expression $\frac{\partial(Q,P)/\partial(q,P)}{\partial(q,p)/\partial(q,P)}$ is ratio of Jacobians.
Evaluate each expression independently by eliminating the repeated variables - then divide them.
In your case, $\frac{\partial(Q,P)/\partial(q,P)}{\partial(q,p)/\partial(q,P)}=\frac{\partial Q/\partial q}{\partial ... | {
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Simultaneity and special relativity Suppose, in inertial reference frame $F_1$, observers A and B are at rest, each having torch, and are separated by some distance and we have put machine M at middle of A and B.
Machine M has light bulbs on both sides ,right and left, so that if it catches light from A which is at lef... | The machine is in reference frame F1, not F2, so It depends who you ask if it's malfunctioning.
Assuming that you know that the box works in this way - then the fact that the machine made a noise only tells you that in F1 - the beams of light reached the machine at the same time.
I can say that the machine is malfuncti... | {
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Different results for the Hamiltonian of a disc rolling on an inclined plane $\hskip2in$
Starting from a Lagrangian of a disc rolling down on a inclined plane without slipping, given by:
$$
\mathcal{L}=\frac{M}{2}\dot{x}^2+\frac{MR^2}{4}\dot{\theta}^2+Mg(x-L)\sin(\alpha) \tag{1}
$$
where $M$ is the disc mass and $\alp... | *
*OP is considering a constrained Lagrangian of the form
$$ L(x,\theta;\dot{x},\dot{\theta};\lambda)~=~\frac{M}{2}\dot{x}^2+\frac{I}{2}\dot{\theta}^2 - V(x) -\lambda(x-R\theta). \tag{A}$$
The 'reduced' Lagrange equation reads
$$ \left(M+\frac{I}{R^2} \right)\ddot{x}~\approx~-V^{\prime}(x). \tag{B}$$
*OP's second 'r... | {
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Question about perturbation theory and even and odd wavefunctions I was solving a question about perturbation theory and I came across something my teacher didn't mention and I can't seem to understand it. In the question there is an external electric field on a H-atom. I can neglect hyperfine structure and lamb shift... | The first order perturbation correction is just the matrix element of the perturbation in respect to the wave function of the state of interest:
$$E^{(1)} = \int dxdydz \psi(x,y,z)^*\hat{V}\psi(x,y,z).$$
The wave function of a bound state can be taken to be real. In an atom some of these functions are even and other ar... | {
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How is it physically possible that the electric field of some charge distributions does not attenuate with the distance? Let's consider for instance an infinite plane sheet of charge: you know that its E-field is vertical and its Absolute value is $\sigma / 2 \epsilon _0$, which is not dependent on the observer positio... | Instead of thinking about the sheet of charge as having infinite dimensions, think of the dimensions being much greater than the distance from the charge where the field is measured. The electric field between the parallel plates of a capacitor is considered uniform (except near the edges) yet the dimensions of the pla... | {
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Two standing sticks with a hinge, seems to lead to a paradox? I have kind of a paradoxal question about two sticks connected through a hinge that rest on a frictionless floor at the bottom and frictionless walls on each side at the top.
What are the forces in the hinges if the total weight of the sticks is equal to 2*F... | The horizontal (normal) forces at the top are equal but opposite. The weight is supported by the normal forces at the bottom. The sum of the torques about the hinge is zero. The system is in equilibrium.
| {
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Is my friend right about omitting $c^2$ in world famous tiny equation? I know $E = mc^2$ says that inertial mass of a system is equal to the total energy content of a system in its rest frame. My friend told me the $c^2$ can be omitted from this equation because that's just an `artifact' when measuring inertia and ener... | This is basically a philosophical question, but I'm going to take what will probably be an unpopular position that your friend's reason is basically wrong in the context of an introduction to special relativity.
Sure, you can work in units where c = 1, and then the equation $E = m c^2$ reduces to $E = m$. But that fact... | {
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How do you combine two spatial modes of light into one spatial mode in optics? Is there a lossless way to combine light coming from two different single photon sources into one spatial mode? Either free space of fiber would be fine.
Let's assume the wavelength and polarization are the same in both input spatial modes.
... | If the two inputs are not in phase, the answer is "no". If they are in phase, they can enter in orthogonal linear polarizations, and be combined to form a single mode at 45 degrees polarization. But if they are not in phase, the resultant polarization will vary randomly so the resultant will be mixed mode.
| {
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Projective representation of $\mathbb{Z}_N\times\mathbb{Z}_N$ by $su(N)$ generators [The same question has been posted in MathStackExchange.]
The projective representation (rep.) of $\mathbb{Z}_N\times\mathbb{Z}_N$ is $\mathbb{Z}_N$-classified.
It can be understood by embedding into a $SU(N)$ representation.
For exam... | I think what you're looking for is closely related to the matrices described in
Patera, J. and Zassenhaus, H., 1988. The Pauli matrices in n
dimensions and finest gradings of simple Lie algebras of type $A_{n− 1}$.
Journal of mathematical physics, 29(3), pp.665-673.
For $\mathfrak{su}(2)$, these are the regular P... | {
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How does surface tension balance the weight of small objects? My understanding of surface tension is that it acts perpendicular to an imaginary line on the surface of the liquid. i.e for a paper clip it will act horizontally, and in the same plane as the surface of water.
Gravity, however will act downwards, and the t... | It is similar to a membrane with a weight in the center. The weight will force a displacement down. The elastic membrane must increase its area to the new configuration.
| {
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What is the physical interpretation of the Lorenz gauge condition? If we integrate both sides of the Lorenz gauge condition, $\nabla \cdot \mathbf{A} = -\frac{1}{c^2}\frac{\partial \phi}{\partial t}$, over a small volume (free of charges for simplicity), we get:
$$
\int_V \nabla \cdot \mathbf{A} \, dV = -\frac{1}{c^2} ... | May I offer perspective on the practical side of things?
Since gauge transformations are ultimately just changes in our description of the system which don't alter the physics, one of our primary purposes for choosing a gauge is ease of calculation.
The Lorenz gauge condition is, importantly, a Lorentz invariant gauge... | {
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On particle diffraction and its relation to the statistical interpretation of the wave function Particles can be diffracted due to their quantum nature and that is understood by their wave-like behavior. Clearly seen in e.g. plane wave solutions of the Schrodinger equation or a superposition of states which can be seen... | IMO you would likely be ale to apply the de Broglie wavelength into a Feynman path integral, allowed paths would be "n" (n is integer) times the de Brogle length. Diffraction is a 2 step process 1) the particle interacts with the EM field of the slit edge material 2) the particle must interact with the screen in a qua... | {
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Why isnt the water from bottle dropping in the tub(volume of bottle/tub doesnt matter) until the level equalises? I can't seem to understand why water in the bottle wont drop down until the water level equalizes so pressure is same in any horizontal plane. As you can see just at tip of bottle,the pressure inside bottle... | If the bottle is closed above, the total pressure due to the weight of the water of the bottle is $\rho gh$ at the tip.
There is much more than 1000 m of air on the water surface. Even air density being about 1/1000 of the water at the sea level, it is enough to weight much more than the water inside.
By experience, i... | {
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Can a fan cause a liquid to cool below room temperature? I enjoy the occasional hot drink, but place it below a small fan in order to cool it to a drinkable temperature. Unfortunately, as expected, I commonly forget about my drink, and it ends up very cold.
In fact, it ends up so cold that it feels much colder than I w... | I don't see how it can be cooled below room temperature.
But if the environment is dry and large enough, the fan can remove molecules from the liquid, increasing the water vapor content in the air. In that process, the air may cool, transfering energy to evaporate the water.
Eventually the room can be cooled. As a co... | {
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Is it right interpretation to interpret $E=mc^2$ using potential energy? I am wondering that it is the right interpretation to interpret $E=mc^2$ with potential energy.
What I mean is this: When I studied nuclear fusion, there was missing mass. The hydrogen's nuclear fusion happens when four hydrogen nucleus fuse into... | @Umaxo is absolutely correct, but I think your problem comes from the fact that you assume that four hydrogen nuclei fuse into a helium nucleus. But, in fact what actually happens in stars like the sun is:
A deuterium nuclei (which is hydrogen with a neutron) and a hydrogen nuclei (just a proton), fuse into Helium-3 (... | {
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How does $g$-force relate to time dilation? In the imdb.com goofs page of the movie Interstellar, I found this statement:
How does time dilation affect $g$-force? If this could be explained in Laymen's terms it would be much appreciated.
| The radial coordinate for the innermost stable prograde orbit at which the planet could orbit the black hole would in natural units of $\rm G=M=c=1$ be at
$$\rm r = 3 + Z_2 - \sqrt{(3-Z_1)(3+Z_1+2Z_2)}$$
with the terms
$$\rm Z_1 = 1 + \sqrt[3]{1-a^2} \left( \sqrt[3]{1+a} + \sqrt[3]{1-a} \right) \ , \ \ Z_2 = \sqrt{3a^2... | {
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Representations of a group In Griffiths' Introduction to Elementary Particles (2ed), at the end of Sec 4.1, he says that
an ordinary scalar belongs to the one-dimensional representation of the rotation group, $SO(3)$, and a vector belongs to the three-dimensional representation; four-vectors belong to the four-dimensi... | Scalar, vectors etc are indeed defined with respect to a group operation (here, $SO(3)$) and the dimensionality of the representation in some cases is enough to identify the representation itself.
It is possible to have representations of $SO(3)$ of dimension $2L+1$. You can just take the states with angular momentum ... | {
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Symmetries of $\mathcal{N}=4$ SYM I know that in $\mathcal{N}=4$ SYM there is superconformal symmetry $su(2,2|4)$.
Does this exhaust all symmetries of this theory?
Or there are some more unobvious symmetries?
Maybe AdS/CFT predict such symmetries?
| Besides $SU(2,2|4)$ superconformal symmetry the ${\cal N}=4$ SYM theory also enjoys a $SU(2,2|4)$ dual superconformal symmetry. Together they generate an infinite-dimensional Yangian symmetry.
| {
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Why doesn't water boil in the oven? I put a pot of water in the oven at $\mathrm{500^\circ F}$ ($\mathrm{260^\circ C}$ , $\mathrm{533 K}$). Over time most of the water evaporated away but it never boiled. Why doesn't it boil?
| The water didn't evaporate. It boiled. If you could look closely at the water in the pot in the oven you would see small bubbles rising within the liquid, which would indicate boiling. But you wouldn't necessarily observe what is sometimes referred to as a "rolling boil, i.e., large bubbles rising in the water indicat... | {
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Why is $ \Delta U = mc_v \Delta T$ applicable in non-constant volume processes (ideal gas ) I can completely accept that the internal energy of an ideal gas is the function of the temperature only, namely
$$U = \frac{f}{2} n R T$$
and that we can define 2 quantities ($c_v$ and $ c_p $) that can give the change in the t... | Any two states on the $P-V$ diagram can be connected by a combination of an isothermal (constant temperature) path and an isochoric (constant volume) path. Since, the internal energy of an ideal gas is a state function, so it doesn't matter which path you take from one point to another as long as the starting and the e... | {
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Why do covariantly constant spinors on the circle require periodic boundary conditions? I am asking in the context of a paper of Witten on instability of Kaluza-Klein spacetimes (https://www.sciencedirect.com/science/article/pii/0550321382900074).
The discussion involves applying Witten's positive energy theorem to the... | Buzz is right; I've realized that covariantly constant solutions on $S^1$ are precisely constants, which of course need periodic boundary conditions.
| {
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Spin-Spin Hamiltonian in coupled harmonic oscillator I was reading about identical particles and i came across this example:
Consider two electrons with spin 1/2.
The Hamiltonian for this system is:
$$Η=\frac{p_1^2}{2m}+\frac{p_2^2}{2m}+\frac{1}{2}m\omega x_1^2+\frac{1}{2}m\omega x_2^2+g\vec{S_1}\cdot\vec{S_2}(x_1-x_2... | We can express the Hilbert space of this system as $\mathcal{H}_{tot}= \mathcal{H}_{0}\otimes\mathcal{H}_\frac{1}{2}$, where $\mathcal{H}_0$ is the spinless part of the Hilbert space and $\mathcal{H}_{\frac{1}{2}}$ is the spin part. So the answer to your first question, is yes, we can split it up.
Consider the state $\... | {
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Symmetry factor of certain 1-loop diagrams in $\phi^4$-theory I have to derive a formula for the symmetry factor of the diagrams of the form
in $\phi^4$-theory, where $\phi$ is a real scalar field. By symmetry factor I mean only the number of possible contractions, which lead to the same diagram (without the factor $1... |
Furthermore, we get $(4!/2)^{n/2}$ to connect each pair of external lines to one of the vertices
But if you consider the interchangeability of internal points then shouldn't it be just a factor of 2 for the way of choosing the connection between external line and vertex?
| {
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Newtons Third Law. How is jumping from a pier different to jumping from a boat? This is the absolute basic of a physics and yet 2 hours of googling fails to find an answer ! So ignoring all vertical movement and just concentrating on the horizontal movement:-
*
*A man who weighs 75 kg jumps off a pier (steps off hori... | In the first case the man does a force in the pier and the pier does a force in the man. Both of 150 N. Of course only during the leg's impulse.
In the second case, the man does a force in the boat and the boat does a force in the man. Both of 150 N.
No difference concerning the third law.
And third law is conservation... | {
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Deriving position $\mathbf{r}$ in two-body problem with composite In classical mechanics, we know that accelerations are oppositely directed and inversely proportional to the masses:
$$m_1 \mathbf{a}_1 = -m_2 \mathbf{a}_2.$$
Let's say that we have a three-body system, where none of the masses of the bodies are equal. I... | The two composite bodies, can be reduced down to one body with combined mass, and the location where forces are applied being the barycenter (the center of mass). Then we have the following two body problem.
Consider two free massive bodies located at $\boldsymbol{r}_1$ and $\boldsymbol{r}_2$ at some moment in time. A ... | {
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Ways of calculating coefficient of performance (COP) Calculating COP with the help of:
*
*Power (i.e. a whole plant)
*Enthalpies (with refrigerant)
*Using Carnot's COP as a comparison (this is mainly theoretical).
The refrigerant could be R134a as a processing medium, where it acts as different temperatures on the... | I'm not sure about an actual power plant as a whole (I am not conversant in power plant engineering). But I would guess you would need to apply the general definition of COP. The COP is defined as the ratio of the desired heat transfer to the work required to make the transfer. What is considered to be the "desired hea... | {
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"timestamp": "2023-03-29T00:00:00",
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Young tableaux for conformal group representation This is taken from arXiv:1910.14051, pg 32:
Decomposing this $SO(d+ 1, 1)$ representation into $SO(1, 1)× SO(d)$
representations as in (A.4), we find
$$\square \underset{\operatorname{SO}(1,1) \times
\operatorname{SO}(d)}{\longrightarrow}(\bullet)_{-1}
\oplus(\square... | This follows because the $d$-dimensional (global) conformal group ${\rm Conf}(d)$ is locally isomorphic to the proper Lorentz group $G:=SO(d\!+\!1,1)$ in Minkowski space $$\mathbb{R}^{d+1,1}~\cong~\mathbb{R}^d\times \mathbb{R}^{1,1}, \qquad \mathbb{R}^{1,1} ~\cong \underbrace{\mathbb{R}}_{\ni x^+}\times \underbrace{\ma... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is a projectile trajectory parabolic while that of a satellite elliptic? I understand that the parabolic trajectory is an approximation of a more elliptical trajectory, since acceleration due to gravity is taken to be a constant for a projectile. However I'm intrigued to know that what changes in kinetic and potent... | It is simpler to think in terms of the solution of 1/r two body potentials, these are conic sections.
One application is that a moving particle that is subjected to an inverse square law force like gravity or Coulomb's law will follow a path described by one of the conic section:
Each of the conic sections can be ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why change in Gibbs free energy of mixing is not zero? I can't understand why change in Gibbs free energy must be zero when two ideal gases mix. The differential of $G$ is: $$dG=-SdT + Vdp + \sum_i\mu_idn_i$$ When two ideal gases mix and they are initially at same $T$ and $P$ that is $T_1=T_2=T$ and $P_1=P_2=P$ then th... | You might think Ideal solutions don't react thus having a zero Gibbs free energy. But you are forgetting about the Van der Waals forces.
A•••A B•••B ------> A•••B
Ideal solution form van der waals forces which means the vanderwaals between A•••A & B•••B breaks and new vanderwaals are forming between A•••B. So, th... | {
"language": "en",
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"source": "stackexchange",
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Ideal wire and resistance I am having a bit of trouble understanding what an ideal wire is.
Let's assume there is a positive charge on the positive terminal of the battery and a negative on the other side which will give us the same results. Now electric potential is given as integral of E.dr. Therefore the potential w... | An ideal wire is really a limit case. Consider the limit of a resistor as the resistance goes to zero. The electric field approaches zero, but the voltage across it also approaches zero. As long as the current is limited by other components (such as other resistors), the result will approach that of an ideal wire.
I... | {
"language": "en",
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Pressure at the greatest depths humans have dived to without a suit is many atmospheres but humans not crushed, why? I understand that the pressure can be 50 atmospheres: https://en.wikipedia.org/wiki/Deep_diving.
Okay, I understand also that the pressure is equal all around but if someone was put between two metal pla... | The crushing you describe does not happen, because the diver is breathing air that is at the same pressure as his or her surroundings. This means the pressure inside their body is in equilibrium with the pressure outside their body, and so nothing gets crushed.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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Detecting an antimatter asteroid Think about this hypothetical asteroid made of anti matter. Suppose we plan to land on it. How can we detect whether it is made up of antimatter, or regular matter from a distance?
| If we plan to land on it, it must be fairly near Earth. The solar system is made of matter, not anti-matter. In particular, the Sun gives off a stream of particles, the solar wind. This contains largely electrons, protons, and alpha particles.
An anti-matter asteroid near Earth would be bathed in these particles. They ... | {
"language": "en",
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What is the difference between a force and a net force? I read in Newton's first law, it states that an object will continue to have a constant velocity unless acted upon by a force whilst for other articles, it states "unless acted upon by a net force." Which one is correct? Are they both interchangeable? Is there any... | It means the same thing, but is worded more precisely.
The difference becomes clearer when you consider two forces that are applied to the object at the same time - in the less precise wording, of course there would be an acceleration, because there is a force (and another one for the second force). What really happens... | {
"language": "en",
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What does the uncertainty principle tell us about the harmonic oscillator? For the harmonic oscillator we have $\sigma_x \sigma_p = \hbar(n+1/2) $ and by the uncertainty principle $\sigma_x \sigma_p \geq \frac{\hbar}{2}$.
In one of the exercises I was doing I was asked to comment on this result. This makes me think tha... |
" High energy states have greater uncertainty"
This also implies that the lowest energy state or ground state of any quantum system described by a harmonic oscillator has the lowest uncertainty and the lowest energy
As..
$$E_n =\omega \hbar(n+1/2)$$
For the ground state or the lowest energy state we get
$$E_0 = \fra... | {
"language": "en",
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Protected Majorana Zero modes in Kitaev Chain Kitaev's one-dimensional p-wave superconductor Hamiltonian${}^\dagger$ is
\begin{equation}
{\cal H}_{JW}=-J\sum\limits_i(c_{i}^\dagger c_{i+1} + c_{i+1}^\dagger c_{i} + c_{i}^\dagger c_{i+1}^\dagger + c_{i+1} c_{i} - 2gc_{i}^\dagger c_{i}+g)
\end{equation}
After Fourier... | The particle-hole energy spectrum is symmetric around zero energy.
When $g\to 0$, we have two zero energy levels, corresponding to the Majorana zero modes which are localized far away from each other and separated by a gaped medium. It is not possible to move these levels from zero energy individually (as one needs to... | {
"language": "en",
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Is a capacitor a dipole? A few more questions about understanding dipoles I recently learned about dipoles, according to its definition I was wondering if a capacitor can be considered also as a dipole?
Also I was wondering what is the physical meaning of the dipole moment $\vec{p}=qd$?
And my last question is what is ... | Capacitors are technically dipoles, but the two things are usually studied in opposite approximations. With a capacitor, you want to look at it up close, and calculate the electric field between the two plates, because it stores the most energy, and it's the quantity we want to manipulate.
Now, while a dipole is techni... | {
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What are the dependent and independent variables in Ohm's law? I can't quite wrap my head around Ohm's law. The relationship itself is quite intuitive to me. What I don't understand is when a system has dynamic voltages, currents, and resistances. I don't quite understand which variables are dependent and which are ind... | When first introduced to Ohm's law, the canonical example is a battery with a resistor across it. In this case, one builds up an intuition that voltages are somehow "fixed" or "independent", from which currents are to be inferred. However, this is a misconception.
V=IR is a (mostly) true expression relating the voltage... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can we conclude from Maxwell's wave equation that the speed of light is the same regardless of the state of motion of the observers? I am reading a book titled "Relativity Demystified --- A self-teaching guide by David McMahon".
He explains the derivation of electromagnetic wave equation.
$$
\nabla^2 \, \begin{case... | Without experimental evidence the constancy of the speed of light cannot be concluded. If space contained a medium, the aether, for electromagnetic waves one would expect the velocity of light in wrt to the aether to be constant. The aether theory was disproved by the experiment of Michelson and Morley. That left speci... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
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Can we measure $x$ and $p_x$ simultaneously by measuring $p_y$ and $y$ as well? Suppose our plan is to measure experimentally the position $(x,y)$ in the plane and the momentum $(p_{x}, p_{y})$ of a quantum particle. Assuming the canonical commutation relation between $x$ and $p_{x}$, we will bypass so to speak the law... | In this case, we can consider the state $|\psi\rangle_{xy}$ in the $xy$ plane to be a tensor product of two one-dimensional states, $|\psi\rangle_x$ and $|\psi\rangle_y$. These states are independent: measuring one will not affect the other. Let's conduct your measurements in sequence. First, we have the state
$$
|\psi... | {
"language": "en",
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"source": "stackexchange",
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Time evolution of the standard deviation of an operator How would I find the time evolution of the standard deviation of an operator? For example, how might I find the time evolution $\sigma_x (t)$ of the standard deviation
$\sigma_x = \sqrt{ \langle \hat{x}^2 \rangle - \langle \hat{x} \rangle^2}$
of the position opera... | The standard deviation in question is not an operator. One could find the time evolution of $\langle\hat{x}\rangle$ and $\langle\hat{x}^2\rangle$ and then calculate the time-dependent standard deviation. How one calculates them depends on whether one uses the Schrödinger or the Heisenberg picture.
In Schrödinger pictur... | {
"language": "en",
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What it the physical meaning of kg$^{m}\cdot$m$^{n}\cdot$s$^{l}$ for $m,n,l\in \mathcal{N}$ I have some questions regarding the physical meaning of the units.
*
*The unit of Planck's constant $h$ is J$\cdot$s= kg$\cdot$m$^2\cdot$s$^{-1}$ in the SI system. My question is: When multiplying by m$^2$ what is the meaning ... | Basic interpretation of Plank constant is that it is "uncertainty quanta", which is best seen from uncertainty principle :
$$ \Delta E \Delta t \geq \frac{\hbar}{2} $$
If to be expressed numerically it would be :
$$ \Delta E \Delta t \geq 5.273 \times 10^{-35} ~[Js]$$
Which means that, if you have particle detection ti... | {
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If gases are not ideal at high pressures, why doesn't raising the temperature of a gas make it less ideal? I'm learning about ideal and non-ideal gas behavior in Chemistry, and my textbook and online sources say two things :
*
*Gases behave ideally at high temperatures and low pressures.
*Gases behave non-ideally at... | The reasons for nonideal behavior are different for the two scenarios you describe, and as such they do not cancel each other out.
The ideal gas law is an approximation that works well when the gas molecules are very far apart and don't collide very often. they are close together at low temperatures and high pressures ... | {
"language": "en",
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Literature to learn thermodynamics coming from a statistical physics background I would like to read a book which rigorously defines what in applied thermodynamics is called "heat" and gives a proper derivation of the laws of thermodynamics from quantum statistics and the principle of Von Neumann entropy maximization.
... | Part II of my online book
A. Neumaier and D. Westra, Classical and Quantum Mechanics via Lie algebras (2011 version)
does precisely that. While it starts off with a chapter giving an axiomatic treatment of phenomenological thermodynamic, the next three chapters rederive all formulas (including the three laws of thermod... | {
"language": "en",
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Magnetic levitation using permanent magnets Is it possible to achieve magnetic levitation (rail) using permanent magnets? So no electromagnets or superconductivity. If yes, how? If not, why not?
| Technically speaking, yes, but practically speaking, no.
The levitation system of maglev trains generally only relies on static fields, which hypothetically could be reproduced with permanent magnets. You can even make models using permanent magnets. However, for actual practical systems (at least for human transport),... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/563697",
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"source": "stackexchange",
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Equilibrium reaction forces on an inclined rod
Suppose we have a uniform steel rod leaning against a frictionless wall in static
equilibrium. The frictional force between the lower end and the floor is less than its limiting value by a finite amount. The rod is supplied some amount of heat so that it expands. Assume ... | My approach:
on the wall $\vec R_2 =\begin{pmatrix}R_{2x}\\0\end{pmatrix}$ and on the floor $\vec R_1 =\begin{pmatrix}R_{1x}\\R_{1y}\end{pmatrix}$
sum of forces in x direction:
$R_{2x}-R_{1x}=0$ gives $R_{2x} = R_{1x}$
sum of forces in y direction:
$-mg+R_{1y} = 0$ gives $R_{1y} = mg$
further $R_{2x}=\mu \, R_{1y} = \m... | {
"language": "en",
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Why don't the nuclear fusion processes inside the sun produce electron antineutrinos $(\bar\nu_e)$? Why don't the nuclear fusion processes inside the sun produce $\bar\nu_e$ despite having the same mass as $\nu_e$? Is the reason as simple as "there is no production channel for $\bar\nu_e$s." ?
| The sun starts with electrons and protons and fuses the protons into heavier nuclei. Almost all stable nuclei contain neutrons, which means that in this process, protons have to be converted to neutrons. This requires the destruction of electrons by charge conservation, which in turn requires the creation of electron n... | {
"language": "en",
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Product of an odd number of Dirac $\gamma$ matrices Suppose $n$ is an odd number. Why can we write $a\llap{/}_1 a\llap{/}_2 ... a\llap{/}_n$ as
$$a\llap{/}_1 a\llap{/}_2 ... a\llap{/}_n = V_\mu \gamma^\mu + A_\mu \gamma^\mu \gamma_5$$
for some $V_\mu, A_\mu$?
I know that $\gamma^{\mu} \gamma^\nu=g^{\mu \nu}-i\sigma^{\m... | It follows from the basic anticommutation relation for the $\gamma$-matrices, $\{\gamma_\mu,\gamma_\nu\}=2g_{\mu\nu}$, that (i) two different $\gamma$-matrices anticommute, and that (ii) the square of a single $\gamma$-matrix is plus or minus the unit matrix. Then the statement in the question follows from the followin... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why is internal resistance of battery considered outside the terminals although it is present between the terminals inside the battery In ideal battery the internal resistance is zero whereas in non-ideal battery there is some internal resistance now this internal resistance is due to the battery material (electrolyte)... | For learning about internal resistance we often draw circuits with the (two-stroke) battery symbol next to a resistor representing the internal resistance – and in series with it and the rest of the circuit. In such diagrams the battery symbol does not represent the whole battery! It represents just the emf-giving part... | {
"language": "en",
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What is the effect on charge of a metal when light frequency continues to increase above photoelectric work function? For a given metal, light below a certain frequency will not eject an electron. Once the threshold or minimum energy is met to eject an electron, what happens as light frequency increases past the photoe... |
Do more and more electrons eject from the metal and thus the metal becomes increasingly positively charged?
Yes , more and more electrons will get ejected when you increase the frequency of the light.However, the rate of emission is still the same. Moreover in the experiment the metal is earthed to maintain electric... | {
"language": "en",
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Commuting the time evolution operator Given the time evolution operator $U(t, t_0)$, I don't understand why it is true that for a time-independent operator Q,
$$[Q, U(t, t_{0})] = 0 \Leftrightarrow [Q, H(t)] = 0 $$
where H is the Hamiltonian.
| The unitary time evolution operator is by construction
$$
U(t, t_0) = \text{e}^{-\frac{i}{\hbar} H (t - t_0)}
$$
which is understood as the series expansion of the exponential function!
Therefore one observes that
$$
\text{e}^{-\frac{i}{\hbar} H (t - t_0)} = \sum_{k = 0}^{\infty} \frac{(-\frac{i}{\hbar} H (t - t_0))^k}... | {
"language": "en",
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Correct integration bounds and radius variable for voltage given charge density My professor worked out this problem and I'm confused with the use of radius.
The hemisphere has a radius $a$. The charge density if a function where $\rho = Ar\cos^2(\theta)$ using spherical coordinates. You solve for $q$, with $dq = \rho\... |
well my friend you seem to be committing a very trivial mistake in perceiving the $'r'$ in the charge density and the $'r'$ in the denominator of the potential as the same! so to avoid the confusion use different notation for the problem as described in the figure above !
| {
"language": "en",
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Sanity check: can we define "arbitrary" hamiltonians? I just want to do a sanity check on my understanding of Hamiltonian mechanics:
My understanding is: For any number $n$, take the phase space $\mathbb R^{2n}$, and take any arbitrary differentiable function $H:\mathbb R^{2n}\to \mathbb R$ to be the Hamiltonian. Then ... | Yes, any smooth function on the phase space can be Hamiltonian. And to any Hamiltonian corresponds a Hamiltonian vector field $V_H$, such that
$$
i_{V_H} \omega = -dH
$$
In the simple case of $\mathbb{R}^{2n}$ the symplectic form is
$$
\omega = \sum_i d q_i \wedge d p_i
$$
| {
"language": "en",
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Are there any quantum effects which we can see in every day life? I am wondering if there are any natural phenomenon in every-day life that cannot be explained by classical physics but can only be explained by quantum mechanics. By classical physics, I mean Newtonian mechanics and Maxwell's electromagnetic theory.
I kn... | I think Fluorescence could be an answer. You only see light beeing emitted at certain frecuencies, correlated to the quantized energy levels in atoms.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/565248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is Maxwell's theory of electromagnetism completely validated? Most of the time, I read articles or watch videos saying that Hertz experiments validated Maxwell's theory of electromagnetism. But Hertz only confirmed the existence of waves (that are perhaps electromagnetic) and that they travel at the speed of light.
He ... | It is easy to show, using the Lorentz force, that the work done per unit of volume and time on free charges by EM plane waves is $W = \mathbf {E.J}$, where $\mathbf E$ is the electric field of the wave and $\mathbf J$ is the density of current. The magnetic field contribution for the Lorentz force cancels out, and does... | {
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Expected Gaussian distribution on screen in Stern-Gerlach Experiment In the Stern-Gerlach experiment, they used a beam of silver atoms travelling in z direction (which has valence electron in the 5s shell), so the electrons carry zero angular momentum. Now since they have zero angular momentum, their magnetic moment mu... | Stern and Gerlach didn't know that electrons carry spin, so they did in fact think that their intrinsic magnetic moment is 0. But through experiments they still knew that the magnetic moment of a silver atom was non-zero. They attributed this to an orbital angular momentum with quantum number $l=1$, though, so they exp... | {
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Does "adding a constant to the potential energy" imply "adding a total time derivative to the Lagrangian." I am trying to understand why Taylor says in his Classical Mechanics text, "we can always subtract a constant from the potential without effecting any physics."
I assume "doesn't effect any physics" means the equa... | Yes, indeed the equations of motion are unaltered; clearly we consider potentials of the form $V(\mathbf q)$, Hence, the only contribution to the Euler Lagrange equations done by the potential is in the term $\frac{\partial L}{\partial q}$, since a constant vanishes under a derivative; then adding a constant do not aff... | {
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Conserved quantity for any motion in 1D Newton's second law for a nonrelativistic particle of mass $m$ in 1D, reads, $$F \bigg(x, \frac{\mathrm{d}x}{\mathrm{d} t}, t \bigg)=m \frac{\mathrm{d}^2 x}{\mathrm{d} t^2}$$, where $F$ is the net force function. Now, if we assume that $F$ is time-independent and that $v=\frac{\m... | For what it's worth, one may show that a 2nd-order ODE
$$\ddot{x}~=~f(x,\dot{x})$$
without explicit time-dependence, or equivalently, a pair of 1st-order ODEs of the form
$$ \dot{x}~=~v, \qquad \dot{v}~=~f(x,v),$$
always has a local Hamiltonian formulation, cf. e.g. this Phys.SE post. The Hamiltonian is a conserved qu... | {
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Locally flatness in general relativity My professor made following statement:
The spacetime of GR is curved in the presence of strong gravitational fields. The effects of curvature
manifest themselves at large distances. Locally, one can choose a flat Minkowskian metric.
I dont get it:
I... | As much as the Earth looks flat it is not, however if you take a really small part of it, you can essentially say that it is flat, same is the case with space-time.
On Space-Time, you can define coordinates, which at the given point are flat and follow Minkowskian geometry, but if you start moving away from them, the d... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/566145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Why does a given molecule is able to exclude 4 time of its volume in the correction term of Van der Waal's equation? In the Van der Waal's equation in volume correction term, a given molecule is able to exclude 4 times of its volume from the volume of the container. Why is it so?
From the Wikipedia page on the topic, i... | Suppose you have a container with volume $V$. If the particles were point-like, then each one's center can be anywhere within a volume $V$. On the other hand, if the particles have radius $r$, and one particle has its center at the origin, then a second particle can't have its center within $2r$ of the origin. If it we... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/566468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is the digital representation of acoustic waves directly proportional to displacement(or pressure)? In most audio editing software(like audacity) waveform can be viewed. Zoomed in these look like actual waveforms. Out of curiosity I downloaded some frequency sweep test files(which are intended to test the linearity of ... | An ideal loudspeaker generates pressure fluctuations proportional to the voltage and an ideal microphone generates voltages proportional to pressure fluctations. In practice (for non-ideal transducers), there will be frequency-dependent phase differences and a frequency-depenent amplitude as well.
The energy stored in ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/566635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Why are the number of magnetic field lines finite in a particular area? One can draw/imagine as many unique (curved/straight) lines as he/she wants in some specified finite area (assuming that each line is unique if it doesn't overlap with another line). Then how can the number of field lines in a particular area be a ... |
why there are gaps between the the iron filling lines?
Iron filings are ferromagnetic. They don't just show the field, they change it.
...hence the iron filings align themselves to stronger field lines.
The filings self-organize into distinct lines because their presence concentrates the field. Magnetic field lines... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/566712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "31",
"answer_count": 5,
"answer_id": 3
} |
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