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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Quantum Efficiency Estimation

Might there be a way to do a rough estimate of the quantum efficiency of a photo-detector like a CCD or CMOS sensor based only on a picture taken with it? I've read papers and guides (like this one: ...
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Formula for Electrical Arc Length

I was playing with some High-Voltage the other day, when a question popped into my head. Can you calculate length of an electrical arc? It probably would be proportional to :- 1. Voltage of the source ...
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62 views

In general, could any ad-hoc relationship of constants be useful?

In general; if one creates an ad-hoc relationship of constants, can we use it to solve equations OR is it just an abstract/artificial math construct? I'm a grad student and as we all know, these ...
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45 views

Dimension agreement in canonical transformation

In this Physics.SE post, there is a transformation: $$Q = q,$$ $$P = \sqrt{p} - \sqrt{q}.$$ for Hamiltonian $H = \frac{p^2}{2}$. The post discusses the validity of this transformation as a canonical ...
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why are the anthropometric units (which are about as big as we are) as large as they are relative to their corresponding Planck units? [duplicate]

so this might have some duplicated inquiry that this question or this question had, and while i think i have some of my own opinion about it, i would like to ask the community here for more opinions. ...
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Is it possible to change units in order to simplify the value of an exponential?

I have the equation $$F=e^{E_0 i \pi}, $$ where $E_0$ is the time-independent electric field, and $F$ is just some important value I am trying to calculate. Obviously, it would be better if $F=-1$, ...
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1answer
61 views

$c^4$ in Einstein field equations

I have read many derivations of Einstein field equations (done one myself), but none of them explain why the constant term should have a $c^4$ in the denominator. the $8{\pi}G$ term can be obtained ...
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47 views

Problems with units of entropy in statistical thermodynamics

The statistical thermodynamics definition of entropy: $S = kN \ln (W)$ troubles me a lot with the problem of dimenstions. where $S$ is entropy; $k$, the Boltzmann constant; $N$ the number of particles ...
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121 views

Why are laws of physics always of product forms?

A first observation is that all the extant laws of physics are of product forms. This phenomenon is somewhat intriguing. The question is: why do law of physics always take, instead of a sum of two ...
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78 views

Units of a discrete Fourier transform

Normally a Fourier transform (FT) of a function of one variable is defined as $$f_k=\int^\infty_{-\infty}f(x)\exp\left(-2\pi i k x\right) dx.$$ This means that $f_k$ gets the units of $f$ times the ...
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35 views

Deborah Number for harmonic excitation

I think I do not understand well the concept of Deborah number. It is presented in the sources available to me as the ratio between the relaxation time of a fluid and a characteristic time scale of ...
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1answer
152 views

Find out the dimension of $\frac{a}{b}$ [closed]

$E=b-\frac{x^2}{at}$ [x=distance,t=time, E=energy] I have tried following but don't know whether I am correct or not $$\frac{x^2}{at}=E$$ $$\frac{L^2}{aT}=ML^2T^{-2}$$ $$a=M^{-1}T^{1}$$ ...
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100 views

Is quantity a dimension? [closed]

We believe that time is a dimension and that $x$,$ y$, $z$ are dimensions in space. Is quantity a dimension like these? And if not, how do we have dimensionless numbers (like $e$, $\pi$ etc.)?
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65 views

The position of a particle at any time $t$ is given by $S = V0/a [1-e^{-at}]$. What are the dimensions of $a$ and $V_0$?

To find the dimensions of and V0, I must know the dimension of S and e. So I want to know it.
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1answer
106 views

How to interpret $t^2$? [closed]

I can't think of the meaning of squaring the Time (multiplying it by itself). It makes sense in Mathematics. But how can you figure it out in nature (or physics). As an example, the formula ...
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183 views

Apparent dimensional mismatch after taking derivative

Suppose I have a variable $x$ and a constant $a$, each having the dimension of length. That is $[x]=[a]=[L]$ where square brackets denote the dimension of the physical quantity contained within them. ...
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4answers
292 views

Why isn't it $E \approx 27.642 \times mc^2$?

Sorry for the strange question, but why is it that many of the most important physical equations don't have ugly numbers (i.e., "arbitrary" irrational factors) to line up both sides? Why can so many ...
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2answers
151 views

Taking force, mass and length as base units, find the dimensional formula of velocity [closed]

My doubt is that how can force be considered as a base quantity. Is that possible? How can I represent the dimension of velocity using it?
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44 views

Finding dimensional formula

$$y(x,t)=2A\sin(Kx)\cos(\omega t)$$ $A$ and $x$ are in metre, $\omega$ is angular frequency. Then find dimensions of $A$ and $K$. In this equation how can I find the dimension of $K$?
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68 views

Is dimensional analysis always sufficient to establish equivalence of quantities?

In dealing with the Biot-Savart law, it was argued that $$ q\frac{d\vec{s}}{dt}\equiv Id\vec{s} $$ using the fact that the units are equal. Does this kind of argument always work? It seems too ...
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3answers
102 views

What is a proportionality constant? (Planck's constant)

I understand that Planck's constant is essentially the ratio between the energy of a photon and its frequency. There are 2 things that im trying to verify: isn't the number that Planck's constant ...
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194 views

Exponential or logarithm of a dimensionful quantity?

I have a unit measure, say, seconds, $s$. Furthermore let's say I have a dimensionful quantity $r$ that is measure in seconds, $s$. What is the unit measure of $e^r$? ($1/r$ is in $Hz$.) My question ...
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1answer
70 views

Dimensional analysis

I was recently introduced to dimensional analysis and I wanted good references for learning the ideas behind it and representation of the natural world. I'm a grad student in biology. I don't have ...
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116 views

Why do we set $x^0 = ct$ instead of $x^0 = t$?

When we deal with Special Relativity and we start considering spacetime instead of space and time each at once, we usually see books saying that we consider a space with four coordinate $x^\alpha$ ...
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90 views

What does really mean by- power of a number or an exponential function is dimensionless? [duplicate]

Is power of only a number or an exponential function is dimensionless? If power of any other thing can also be dimensionless then please explain with examples.
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80 views

Dimensional Analysis to Determine a Formula

The kinetic energy of a particle confined to a spherical region with a uniform internal potential depends on its mass, the radius of the sphere, and the Planck constant. An electron, confined to such ...
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29 views

Justifying order of magnitude reasoning

So in the context of a set of notes I am reading about acoustics I get to equation (23) in this paper. Basically it comes down to showing that (note the dots above the a's meaning time derivative!) ...
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145 views

Why dimensionality of the Electric Charge varies with the spacetime dimensions?

The point is: We can find via dimensional analysis that the electric charge dimensionality varies with the dimension of space-time. $$[\text{charge}] = eV^{(3-D)/2}$$(You can see below the way I did ...
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53 views

Order of magnitude estimation for some intriguing questions

The physics TA showed us a few examples in which one can estimate many things from first principles and sound logic based and scaling arguments. This led usually to understanding of why some numbers ...
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67 views

How can Planck units be consistent with conflicting dimensions of mass?

I suspect I'm missing something obvious, but I'm coming up blank. I've gotten pretty comfortable with so-called natural units over the years: in doing quantum mechanics/QFT, it's common to set $c = ...
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2answers
96 views

What is dimensional units/quantity and dimensional state

First, I am not a native English-speaking student so I am not good at physics definitions in English. I participated in the MIT e-learning course on classical physics. The 1st lesson is about 3 ...
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129 views

Units of the Stokes-Einstein rotational diffusion coefficient

The Stokes-Einstein rotational diffusion relation tells us that we can write down a rotational diffusion coefficient for a sphere as: $$D_r \approx \frac{k_B T}{\zeta_f} \approx \frac{k_B T}{(8 \pi ...
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Definition of Ohm in SI basic units in words

One way Wikipedia defines Ohm is (this is also teached in school): $$1\Omega =1{\dfrac {{\mbox{V}}}{{\mbox{A}}}}$$ They describe this definition in words, too: The ohm is defined as a resistance ...
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What justifies dimensional analysis?

Dimensional analysis, and the notion that quantities with different units cannot be equal, is often used to justify very specific arguments, for example, you might use it to argue that a particular ...
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138 views

Is there a physical quantity which is the reciprocal/multiplicative inverse of time?

Is there a physical quantity which is the reciprocal/multiplicative inverse of time? If time =distance/speed what is speed/distance. Please forgive my ignorance if there is a really simple ...
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93 views

Dimensional Analysis : Thermodynamics

I was coming across some notes online for phase transitions. In one of the places, the author has written the Claussius-Clayperon equation in this form, $$ \frac{d(ln P)}{d(ln T)} = \frac{T\Delta ...
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229 views

What prediction led to the vacuum catastrophe?

The disagreement between predicted and measured energy density of the vacuum is one of the great unsolved problems of science. According to this book the predicted energy density was obtained as ...
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753 views

Why does Coulomb's constant have units?

I think of Coulomb's constant as a conversion factor (not sure if this is correct). Kind of like how you would do calculations in kg and then times it by the conversion constant to convert your answer ...
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160 views

What does the decay constant mean?

In my curriculum, the decay constant is "the probability of decay per unit time" To me, this seems non-sensical, as the decay constant can be greater than one, which would imply that a particle has a ...
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323 views

Integrating equations with units

I was looking through an old copy of Barron's AP Physics and found this problem relating to impulse which I was initially confused about how to integrate. Example 6.1 During a collision with a ...
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171 views

Could velocity be taken as fundamental instead of time?

In physics time and length are taken as fundamental in the SI system and, as it seems, in the thinking of physicists. Could one instead take velocity, with c as its unit, together with length as ...
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73 views

Surface gravity for a rotating charged black hole

I have that the surface gravity (at the outer event horizon) for a Kerr-Newman black hole is $$ K_+ = \frac{r_+-r_-}{2(r_+^2+(J/M)^2)} = ...
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Dimensional analysis to estimate order of magnitude of quantities

In the Coursera course From the Big Bang to Dark Energy on several occasions dimensional analysis was used to estimate the scale of quantities. This almost seems like a contradiction in terms to me, ...
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Problems with dimensions when solving an ODE

I'd like to solve the following differential equation: $$\frac{dQ}{dt}=\frac{k_BT}{m}-\frac{\alpha Q}{m}$$ where $Q$ has units of $\text{m}^2\text{s}^{-1}$, $k_B$ is Boltzmann's constant, $T$ is ...
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2answers
370 views

Mass dimension of coupling constants in various dimensions

Just a quick question: Suppose I want to consider QED or YM in 4 dimensions we always say that the coupling constants are dimensionless and that the field then has a specific mass dimension. What ...
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1answer
144 views

Why is it meaningless to speak about changes in a dimensional constant?

Every so often,* we get a question about what would happen should there be a change in a physical constant that contains dimensional information, such as $\hbar$, $c$, $G$, or often "the scale of the ...
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141 views

Confusion With How Dimensions Work

Form what I understand if you have an equation such as: $$v = v_0 + at$$ then the dimensions must match on both sides i.e. $L/T = L/T$ (which is true in this case), but I have seen equations such as ...
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Square bracket notation for dimensions and units: usage and conventions

One of the most useful tools in dimensional analysis is the use of square brackets around some physical quantity $q$ to denote its dimension as $$[q].$$ However, the precise meaning of this symbol ...
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Is the number 1 a unit?

In dimensionless analysis, coefficients of quantities which have the same unit for numerator and denominator are said to be dimensionless. I feel the word dimensionless is actually wrong and should be ...
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Scale-invariant differential operator

For example, the differential operator Laplacian is $$\nabla^2 = \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}.$$ My questions are: Is it scale-invariant? what is ...