Questions tagged [dimensional-analysis]

Dimensional analysis means to obtain results by analyzing the units in question, etc. DO NOT USE THIS TAG if your question is about degrees of freedom or spatial dimensions.

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58 views

Proof for property of proportionality used in deriving physical laws like law of gravitation and coulombs law

$$\text{F} \propto m_1m_2$$ $$\text{F} \propto \frac{1}{r^2}$$ Therefore $$\text{F} \propto \frac{m_1m_2}{r^2}$$ In physics one quantity $\text{F}$ is directly proportional to two other quantities ($...
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Could you have a quartic or square degree Celsius or other degree on a temperature scale raised to any power?

Could you have a quartic or square degree Celsius or other degree on a temperature scale raised to any power? The units of the Stefan-Boltzmann constant are watts per square meter per quartic kelvin, ...
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106 views

Is the "space of physical quantities" a field of transcendence degree $6$ or $7$ over the rationals / real numbers?

Excuse my naive question and please let me explain it: In everyday life we experience 3 spatial "dimensions" + time etc. Usually the 3 dimensions are represented by a coordinate system and ...
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A question regarding dimensional analysis and a "Planck matrix"?

Let: the speed of light in a vacuum, $c$, the gravitational constant, $G$, the reduced Planck constant, $\hat{h}$, the Boltzmann constant, $k_B$ the electric constant, $\epsilon_0$ with dimensions $...
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What are $δ$ and $ε$ in the list of Dirac's five 'fundamental constants', concerning his 'Large number hypothesis'?

From Jean-Philippe Uzan's Varying Constants, Gravitation and Cosmology: Dirac formed five dimensionless ratios among which1 δ ≡ H0ħ/mpc2 ∼ 2h × 10−42 and equation M1 and asked the question of which of ...
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21 views

Scaling difficulty of landing a rocket

I am wondering whether it is more difficult to land (as spaceX does) a rocket of height $k L$ than landing a rocket of height $L$? Is there some scaling law? Or at least is there a way to see in the ...
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String length $\gg$ Planck length?

The potential energy of a string $E_{pot}$ is given by $$E_{pot}=T \times L\tag{1}$$ where $T$ is the string tension and $L$ is the length of the string. The internal kinetic energy $E_{kin}$ of a ...
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126 views

Why is $\exp(-1)$ a good benchmark in many physics? [closed]

In many subjects in Physics (not only phys though), we often encounter a value which gives $\exp(-1)$, such as a time constant in circuitry ($\exp(-t/RC)$, $t=RC$ gives $exp(-1)$), lifetime, mean free ...
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75 views

$G$ expressed as a function of $c$?

The escape velocity for a given mass $M$ and a given radius $r$ is given by: $v_e = \sqrt{ \frac{2 G M}{r} }$ With $M$ = 25 kg and $r$ = 1/$c$ we have: $v_e = 1.0001917061 \approx 1$ Can we express $G$...
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36 views

Why is the charge density of a rod in $\rm C/m$ and not in $\rm C/m^2$?

Is it because we consider the rod to be a 1-dimensional straight line? Or is there any other reason? I know it sounds like a stupid question but I am confused in this regard.
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Under what circumstances can we make sense of functions of dimensionful arguments?

Solving Laplace's equation in 2D assuming symmetry $\theta \rightarrow \theta + \delta \theta$ yields $$\phi(r) = \alpha \log(r) + \beta.$$ This is discomforting because $r$ has dimensions of length; ...
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Units in the multipole expansion

In the Wikipedia page (https://en.wikipedia.org/wiki/Multipole_expansion#Expansion_in_Cartesian_coordinates) the expression for the multipole expansion gives the dipole term as $$V(r)=\frac{1}{4\pi\...
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What are units of quantum field (or creation and annihilation) operators?

Let us have a free particle Hamiltonian given as $$ \hat{H} = \sum_{k}\mathcal{E}_k c_k^\dagger c_k \quad ; \quad \mathcal{E}_k=\frac{\hbar^2k^2}{2m} $$ unit of $\hbar$ are $J\cdot s$, the units of ...
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Why does the ladder operator contain the $\hbar ω$ part?

For homework I was deriving the ground state wavefunction for quantum harmonic oscillator using the ladder operator method. From what I've understood, the whole idea starts by factorizing the ...
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Units of Linear Speed in a Circular Motion

During Uniform Circular Motion, the linear speed of the particle is defined as the radius times the angular speed. $$ v = r\omega $$ The units of linear speed is meters/second (m/s). But the units of $...
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Using dimensional analysis to derive formulae for quantities depending on over three factors

Apparently, you can't use dimensional analysis to derive the formulae of quantities which depend on more than three other, different, composite quantities. Why is this so? The argument I've seen for ...
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126 views

What is an accidental symmetry?

Wikipedia describes an accidental symmetry as a symmetry which is present in a renormalizable theory only because the terms which break it have too high a dimension to appear in the Lagrangian but I ...
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Does Planck mass formula demonstrate a quantum theory of gravity?

Planck mass is equal to the square root of h-bar times G, divided by c. Replace G in the gravitational equation with planck mass, h-bar and c and then you result in a quantized equation for gravity. ...
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85 views

Can monochromatic light have units of per frequency?

Plank outlines (page 209, equations 302-304) in his book that a monochromatic ray of frequency $v$ is has intensity of \begin{equation} I_{\nu} = \frac{2 h \nu^{3} F \Omega}{c^{2}} \left( e^{\frac{h\...
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What's the meaning of unit of moment $\mathrm{N\cdot m}$? How do you explain it? [duplicate]

I know the meaning of units with division like $\mathrm{m/s}$ or $\mathrm{m/s^2}$ etc. they make sense, like $2\ \mathrm{m/s}$ is like the car pass $2\ \mathrm m$ in $1$ second, you know what I mean? ...
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What are the units used in $\nabla^a F_{ab}=-4\pi j_b$?

I'm studying Wald's general relativity, in it Wald mentions that the Maxwell's equation is equivalent to $dF=0$ and $\nabla^a F_{ab}=-4\pi j_b$, where $F$ is the Maxwell two form. Wald doesn't ...
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73 views

Buckingham Pi theorem: Why do we need dimentionless parameters?

I've been trying to understand Buckinghams Pi theorem for my lab work on rolling cylinders. I don't exactly get why you need dimensionless parameters? I do understand that having a (not sure of the ...
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What is the dimension of the $A_\mu$ field?

Just as the title say: what is the dimension of the $A_\mu$ field? (If that's of any importance, I'm interested in the dimension of $A$ to figure out the dimension of the coupling constant $g$) I have ...
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How do I use dimensional analysis to find the ratio of potentials at the center and corner of a uniformly charged cube?

The problem goes like this, from Purcell's electromagnetism: Consider a charge distribution that has the constant density $ρ$ everywhere inside a cube of edge $b$ and is zero everywhere outside that ...
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50 views

Unit of angular velocity in oscillatory motion

Suppose an angular oscillatory motion. In the function below, $\alpha$ and $\alpha_o$ are angles measured in radian, $\omega$ is circular frequency ($2\pi/T$) measured in [radian/s] and $t$ is time. $$...
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Test tank physics

In a test tank scaled-down simulation of, for example, a ship stability problem, is it not incorrect to assume that water will behave in a scaled-down fashion with regard to wave period? Don't we need ...
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Units in Bohr's model of an atom

In the Bohr model the values of the orbital kinetic moment and of the allowed radii are quantized according to : $$mv_nr_n = n \hbar, \, r_n = \frac{n^2 \hbar^2}{me^2}$$ By combining the two, i get : $...
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Is the mass of elementary particles proportional to the Planck constant?

I’m reading Manton&Sutcliffe’s “Topological Solitons”. In that book on p. 2, they argue as follows: In a Lorentz invariant theory, and in units where the speed of light is unity, the energy of a ...
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52 views

Dimensionalized heat equation

The heat equation says that $$\rho c \frac{\partial u}{\partial t}= \kappa \frac{\partial^2 u}{\partial x^2}~? $$ How could one devise a nondestructive experiment to say, find the thermal diffusivity ...
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What is the meaning of power laws in physics?

I know the basic meaning of a power law. Scale invariance. Also that with negative exponents, bigger events occur more rarely, like in say earthquakes. But I can't seem to find more physical ...
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74 views

During a dimensional analysis, why is it not allowed to take the absolute value? [closed]

If I know that a formula depends only on two quantities $l$ and $t$ which are in "independent" units (say length and time), why can I only look for an expression of the form $Cm^\alpha t^\...
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How to check the conformal prefactor in a correlation function?

In CFT it is usual practice to extract the so-called conformal prefactor from the correlators in order to isolate a function which depends only on the cross-ratios. For example the $4$-point function ...
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Mathematical Reasoning In Physics

Does this form of reasoning have a name. I often see it but am a little confused on how to read/understand it and wanted to look more into it but don't know what to call it Ex. Let's call the force ...
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How do I non-dimensionalize Newton's Law of Gravitation for the 3-body problem?

I'm attempting to numerically solve the 3-body problem. Using Newton's second law, I've derived a system of 6 second order differential equations, the first three being: $$ m_1\frac{d^2x_1}{dt^2} = -G ...
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What are the physical dimensions of the photonic plane wave operators in the multimode regime?

From this question, it appears that the creation/annihilation operators have units "$1$", i.e. they are of unit dimensionality. However, in the multimode regime, the commutator of two ...
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1answer
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Dimensionless numbers plots

I've used the Buckingham Pi Theorem in an electric water heater to find the relationship of the generated heat with other physical quantities. Parameters: \begin{...
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Why Does this Expression for a Probability Have Units?

In this paper titled "Measurement of Photon Statistics with Live Photoreceptor Cells" the equation 3 which expresses a relation for the probability of a given photocurrent given a certain ...
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1answer
62 views

Unitarity in axion coupling to photons

I have been investigating about axion coupling to EM field, and I have found that it is usually described through the interaction Lagrangian: $$ \mathcal{L} = \frac{1}{4 M } a \tilde{F}_{\mu \nu} F^{\...
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Fundamental quantities in physics

I observed that the fundamental units like meter, kilogram, ampere, kelvin and candela are all indirectly dependent on one single fundamental value 'second' and each other. For example: Meter:- 1 ...
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Interaction of particles with a spectrum of waves, relativistic equation of motion

I have to work with a poorly documented code, that simulates the interaction of particles with a spectrum of waves. The previous author has made the equations of motion dimensionless by using the ...
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1answer
57 views

Dimension of mass of black hole in natural unit

I am reading articles on superradiance, specifically, massive scalr field in Kerr geometry. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.22.2323 One assumption of the article is $$\mu M\ll1,$...
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Confusion on the renormalizability and the dimension of coupling constant (From Srednicki's book)

I am trying to understand the renormalizability and the dimension of couplings from section 18 of Srednicki's QFT book. In section 18, it mentions if we can use finitely many new terms (counter-term) ...
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50 views

How to yield the relativistic equations of motion for a particle in a plasma non-dimensionless

The relativistic equations equations of motion for the ions are: $$ \frac{d\vec{p}}{dt} \ = \ q (\vec{E}+ \vec{u} \times \vec{B}), \ \vec{p} \ = \ m \gamma\vec{v} $$ In many papers discussing test ...
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Is it possible for a valid equation to have different dimension in both sides? [duplicate]

Suppose we have the following equation: $$ \frac{1}{r}\mathrm{d}r=\frac{1}{T}\mathrm{d}T $$ where $r$ be the distance and $\dim r=L^1$, and T can be the temperature with $\dim T =\Theta^1$. In this ...
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Contradiction of Units in Polytropic Process

A polytropic process is a process that obeys the relation $pV^n=C.$ However, when I try to solve a problem involving this relationship with, for example, $n = 1.5$ my use of units in my equations ...
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Are inverses of fundamental SI units also physical units?

I know that Hertz (1/s) is a physical unit, but are there physical units for 1/m or 1/kg too? For example, in the formula for resolvance of a diffraction grating, $\frac{\lambda}{\Delta \lambda} = mN$,...
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Planck constant: photon energy level and density question

$$E = hf = \frac {hc}{\lambda} = J = eV$$ Wikipedia says a photon with $\lambda = 532$ nm would have $E = 2.33$ eV $= 2.33$ J. This calculates for a single photon, so is photon density required to ...
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31 views

$\rm kph/MeV$ for Light yield?

I was reading an article on Scintillation and I came across a peculiar unit $\rm kph/MeV$ for Light yield. It stated for Organic Scintillators, it has a Lower light yield (1-10 kph/MeV). Here do kph ...
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54 views

Why are derived quantities always expressed as a product or division of base quantities? [closed]

I have seen many equations that relate derived quantities with base quantities as a product of those quantities or division of them, e.g., $F=ma$, $V=\frac{S}{t}$, etc, but I want the feel of this ...
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Is there any reason that the Schwarzschild radius of a 5(+)-dimensional object would require different math?

$$R_g = \frac{2GM}{c^2}$$ It doesn't seem to me that the gravitational constant or speed of light particularly care how many dimensions there are in this context. Mass is mass whether it comes in ...

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