Given are two equations:
$$S_1 = ax^2+2hxy+by^2 + c=0$$
$$S_2 = a'x^2+2h'xy+b'y^2 + c'=0$$
This source states that there are several methods to solve for ##x## and ##y##. One of them is the following quote:"Treat equation ##S_1## as a quadratic equation in ##x## and solve it for ##x## in terms...
Some rotational quantum states are not allowed for a rotating particle. At quantum level, these "forbidden" quantum states is based on the requirement of the total wavefunction being either symmetrical or anti-symmetrical, depending on whether the particle is a fermion or boson. The particle's...
I'm trying to understand the detailed concept of why the density of states formula is accurate enough to calculate the number of quantum states of an energy level, including degeneracy, within a small energy interval of ##dE##.
The discrete energie levels are calculated by
$$E = \frac{h^2 \cdot...
It appears that the most probable energy level according to the Maxwell-Boltzmann distribution is not equal to the most probable speed squared multiplied by ##\frac{1}{2}m##. The most probable speed has a different value.
$$E_{max} = \frac{k_BT}{2}$$
$$v_{max} = \sqrt{\frac{2k_BT}{m}}$$
I am...
I probably haven’t thought this through. A sideview of a closed container filled with air consisting of two vertical cylinders (with radius ##r_1## and ##r_2##) are connected by two horizontal tubes. The container is separated by a small and a large lid (red) that are circular and can move up...
Summary: Integrating the 1 dimensional MB Distribution in terms of translational kinetic energy up to infinity, does not yield ##\frac{1}{2}k_BT## as it should be.
The format for the 3 dimensional Maxwell-Boltzmann Distribution is ##A\cdot e^{-\frac{E}{k_BT}} \cdot g(E)## in which ##A## can be...
I delved a bit into the kinetic theory of gases and it got me wondered how it is discovered that the temperature, and thus heat capacity, is dependent on the number of degrees of freedom of a molecule or atom.
I know that from the piston experiment a certain constant value can be found for the...
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The relationship between entropy ##S##, the total number of particles ##N##, the total energy ##U(β)##, the partition function ##Z(β## and a yet to be defined constant ##β## is:
$$S(\beta)=k_BN \cdot \ln(Z(\beta)) - \beta k_B \cdot U(\beta)$$
Which leads to:
$$\frac{dS}{d\beta} =...
I have been reading about the quantum effects that limit the Maxwell-Boltzmann Distribution under certain conditions which leads to the Bose-Einstein or Fermi-Dirac Distribution.
I have difficulty grasping the reasons why these quantum-effects occur only at certain conditions and why exactly...
I've seen a simple guessing show but I'm a bit surprised by its probability results and don't know what I'm doing wrong.
A prize is hidden in 1 of 7 boxes and a person has to guess in which box it is.
He opens a box one after the other and I questioned myself what the probability might be for...
A question came up about deducing the number of possible energy states within a certain momentum ##p## using momentum space.
To make my question easier to understand, I deliberately chose ##p## and not a particular increment ##dp## and I assume a 2 dimensional momentum space with coordinates...
Hello,
I was watching a video on the derivation of the Maxwell-Boltzmann distribution function which would eventually lead to:
$$\frac{e^{-\beta \cdot \epsilon_i}}{\sum_{i=0}^n e^{-\beta \cdot \epsilon_i}}$$
To do this, initially, the number of possible permutations ##\Omega## of a total of...
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I have a question regarding calculating the total reached height ##H## of an object that is thrown upwards with an initial velocity ##v_i##, while taking into account the decreasing gravitational deceleration ##g## with height as well.The formula would be something like:
$$v_i \cdot t...
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I have been trying to solve this problem but I can't seem to find a way.
Given are ##n## cards and each card can show one of two values: M or K.
How many possible permutations are there in which there are as many cards with M as there are with K? Given that ##n## is an even amount of...
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When reading a bit about substractive color mixing, a question came up.
I understood that when mixing Cyan, Magenta and Yellow paint, you would get black because all these 3 colors that get reflected by each corresponding paint are absorbed by the other paint in the mix, leaving no...
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If I understand correctly, according to Kepler, a planet orbiting a star should cover a constant surface area of its ellipsoid or circular orbit per time unit, regardless of the distance of that planet to the star.
I have tried to prove this by using a scenario in which a planet is...
I’ve been wachting a video here on interference maxima when light passes a grating and 3 questions came up.
1.
The lecturer shows the following picture:
He mentions that each purple line from each slit (distance ##d## apart) that meet at the same point on the right side (for constructive...
Homework Statement
Two vehicles, A and B, are approaching each other. Vehicle A has a constant velocity of ##v_A##. Vehicle B has a constant acceleration ##a## with an initial velocity of ##v_B##. The starting distance at ##t=0## between them is ##D##.
Is there a way to solve at what time...
I was looking at the following graph showing the relationship between redshift and distance for a constant, accelerating and decelerating expansion of the universe.
Source
Looking at the accelerating expansion line (red), I tried to reason why it would show a line that deviates upwards from...
I have some more detailed questions on this further on but I found it better to start with a very basic question first:
If higher temperature is correlated with shorter emitting wavelengths, how come there are incandescent light bulbs that emit yellowish light but are hotter than incandescent...
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I noticed something peculiar when looking at the Minkowski Diagram and I'm not sure how to interpret this.
Let observer A be the reference frame for the diagram and B someone who travels with respect to A.
The red line is the displacement of light over time. The blue, yellow and green...
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I did a calculation to determine whether a liquid with a fixed volume ##V##, would be spread over a larger surface area ##A## on the inside mantle of a cylinder, if the cylinder has a larger radius ##r##. So I’d like to find a relationship between the radius ##r## and the area ##A## over...
I’ve been reading on the physics of how a Magnetic Resonance Imaging (MRI) works from this link regarding the T1 and T2 relaxation time of protons.
So far I understand the concept except what they’re trying to explain in the last 2 paragraphs about spin echo.
This is what I got from these last...
Hello,
I have a small confusion regarding the formula that calculates the luminous flux of a light source, which is given by:
(See this Wiki)
The ##\bar y (λ)## is the so-called luminosity function that corrects for the sensitivity of the human eye for a particular chosen wavelength.
The...
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I have been following lectures found here on the end of the lifecycle of stars (while trying my best not to get distracted by his bow tie) and I have some questions on this. Some of them being general questions and some a bit more detailed.
1. First a more general quesion: I’ve read...
I’m probably missing something obvious here but I can’t figure out what.
Consider a galaxy, starting at an initial distance of ##D_0##, recessing with a constant velocity. After a time Δt, we’d measure a larger distance ##D_{t}##. From this scenario I’d conclude that the Hubble value based on...
I have a question about the velocity of a mass that is being accelerated from its resting position, for deriving its energy according to ##E=mc^2##
The energy needed for accelerating a mass is: ##E = F \cdot s = m \cdot a \cdot s##. Where s is the distance over which the mass is moved.
Now, I...
I'm trying to deduce the weight of 1 atomic mass unit (##1u##) in ##kilograms## from the following scenario:
One atomic mass unit is 1/12 of the weight of a ##^{12}C## atom in its ground state. A ##^{12}C## atom consists of 6 protons, 6 neutrons and 6 electrons. This means that
$$1u =...
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I’ve been watching a lecture about how in astronomy one would be able to calculate the radius of 2 stars by measuring the velocity of the orbit and then measure the time how long the luminosity of a star dims when one star is behind the other.
After thinking this a bit through, I came...
I had a question regarding calculating the area of a circular cap on a sphere. From what I’ve read, the area should be calculated according to;
$$A = 2πr^2 \cdot (1 – cos (\frac{θ}{2} )$$
However, I have another way but I don’t understand why this isn’t correct.
The circular area can be...