The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass per unit volume. The symbol most often used for density is ρ (the lower case Greek letter rho), although the Latin letter D can also be used. Mathematically, density is defined as mass divided by volume:
ρ
=
m
V
{\displaystyle \rho ={\frac {m}{V}}}
where ρ is the density, m is the mass, and V is the volume. In some cases (for instance, in the United States oil and gas industry), density is loosely defined as its weight per unit volume, although this is scientifically inaccurate – this quantity is more specifically called specific weight.
For a pure substance the density has the same numerical value as its mass concentration.
Different materials usually have different densities, and density may be relevant to buoyancy, purity and packaging. Osmium and iridium are the densest known elements at standard conditions for temperature and pressure.
To simplify comparisons of density across different systems of units, it is sometimes replaced by the dimensionless quantity "relative density" or "specific gravity", i.e. the ratio of the density of the material to that of a standard material, usually water. Thus a relative density less than one relative to water means that the substance floats in water.
The density of a material varies with temperature and pressure. This variation is typically small for solids and liquids but much greater for gases. Increasing the pressure on an object decreases the volume of the object and thus increases its density. Increasing the temperature of a substance (with a few exceptions) decreases its density by increasing its volume. In most materials, heating the bottom of a fluid results in convection of the heat from the bottom to the top, due to the decrease in the density of the heated fluid. This causes it to rise relative to more dense unheated material.
The reciprocal of the density of a substance is occasionally called its specific volume, a term sometimes used in thermodynamics. Density is an intensive property in that increasing the amount of a substance does not increase its density; rather it increases its mass.
Hi there, I've worked through most of this question but I'm stuck on the final part, showing that total bulk current ##I_B## is equal and opposite to total surface current ##I_S##. I calculated ##\vec H## the normal way I would if I was looking for ##\vec B## in an infinitely long cylindrical...
Summary:: Given statement: according to one model of the sun, the central mass density is 1.53x10^5kg.m^-3 and the mean opacity at the center is 0.217m^2kg^-1
Given statement: according to one model of the sun, the central mass density is 1.53x10^5kg.m^-3 and the mean opacity at the center is...
Hello! I was reading up on methods for determining the density of the universe and I came across this page: https://hypertextbook.com/facts/2000/ChristinaCheng.shtml
I tried using equation stated, Ω=(2/3Λ)(c^2/H^2), with SI unit versions of both variables:
Λ=1.1056 * 10^-52 m^-2
H=2.1927 *...
The answer is that the charge density would be -σ, I cannot for the life of me understand why would that be the case. Of course it makes sense but I can't convince myself that it would be the only possible answer.
I have tried to apply Gauss law a few times, but it doesn't yield anything.
Electric field for the semi-circle
$$E = - \frac {πKλ} {2R} $$
In this case E is equals to 10 N/C
Electric field for the straighten wire
$$E = 2Kλ * ( 1 - \frac {2y} {\sqrt{4y^2 + L^2}})$$
In this case E is equals to 8 N/C
What I'm searching is R, λ, and the length of the wire, so I think...
I've been think about it for hours but I'm really out of clue here... The only things I could think of are obvious or useless... Any help would be greatly appreciated.
I read that density matrices are useful in Physics mainly to describe a) mixtures; we do not know the wave function of the system so ##\psi## is random and b) entangled systems. I'd like to focus on the later.
Let us have a system ##S_1## entangled to the system ##S_2##. Thus we start from a...
UNITS
m is meters
kg is kiliograms
K is degrees Kelvin
s is seconds
J is joules
u is daltons = 1.66053906660(50)×10−27 kg
https://en.wikipedia.org/wiki/Dalton_(unit)
1 pc = 3.085678 x 1016 m
CONSTANTS
MH = mass of hydrogen atom = 1.007825 u
= 1.673532784796145 ×10−27 kg...
I'd like to show that, by minimizing this functional
$$\Omega[\hat \rho] = \text{Tr} \hat \rho \left[ \hat H - \mu \hat N + \frac 1 {\beta} \log \hat \rho \right]$$
I get the well known expression
$$\Omega[\hat \rho_0] = - \frac 1 {\beta} \log \text{Tr} e^{-\beta (\hat H - \mu \hat N )}$$
I'm...
2 scenarios:
1) Situation in the movie "2012", where volume of ocean water hasn't changed, but worldwide seismic activity has caused massive tidal waves. In one scene, a ship is sailing by Mt. Everest close to its peak. In such a scenario, I believe the air density outside the ship would the...
I'm not sure I understand why I need to use ##d##.. Maybe they want me to have the potential be zero at ##A##?
In any case, I have found$$V(B)=\alpha k\int_0^L\frac{x}{\sqrt{b^2+\left(x-\frac{L}{2}\right)^2}}dx+C=\frac{\alpha...
Homework Statement:: The relative density of water is determined by the rate at which it expands (and contracts) with changes in temperature. At approximately what other temperature T does water have the same density as at 1 ∘C ?
Relevant Equations:: Just looking at graphical and analyzing...
I attached a drawing of the problem for a better understanding and my attempted solutions.
The first point is fairly simple but there's something that I can't figure out.
dq=λdx=kxdx
Q=∫ k x dx from 0 to L -> Q=k[x^2/2]0-L -> Q=(L^2/2)k -> k=2Q/L^2
This is what I came up with. I integrated...
Question 1;
a) P=E/t
E=5.796*10^7 J energy produced per day during the summer
However, I am not certain how to calculate the time period, since although this concerns the energy produced per day, the sun does not shine for the entire duration of this 24 hour period. Also, I am unsure of the...
Answer : Using Pascal's law, this is my answer : ##\color{blue}{\boxed{\vec F_a = \vec F_c < \vec F_b}}##.
Reasoning :
Forces ##F_a## and ##F_c## are equal because the pressures required at the two cylinders for case (c) is the same as that required in (a). It doesn't matter how many of those...
I'm modelling the interiors of core-dominated (exo)planets. The EoS I use in my calculations are mostly either a Birch-Murnaghan formulation or a Mie-Grüneisen-Debye formulation. In either case, the ambient density ρ0 at ambient pressure and temperature are required for the implementation...
I am trying to understand an excerpt from an article describing the vibrations of a string (eg. guitar/piano) which reads as follows:
This is basically the wave equation with Δm representing a small piece of mass from an interval of the string and two forces added to the right side.
He...
There are some universe models where ##\Lambda < 0##. In this case, the energy density of the dark-energy becomes negative. At this point, does it make sense to talk about "negative dark energy density"? Or is it possible to think of this energy as curvature on space-time? Such that, ##\Lambda <...
I got the correct answer to this question with the following calculations, but I do need some correction in terms of what units I'm integrating across.
ρ##\rho ## is density.
mtot=n##m_{tot}=nM##, where n is the number of moles of a substance and M is the molar mass of the substance...
I have a question related to linearity of power spectral density calculation.
Suppose I have a time series, divided into some epochs. If I compute PSD by Welch's method with a time window equal to the length of an epoch and without any overlap, I obtain this result:
If I calculate the...
I am reading an article, which talks about graduated dark energy (gDE) model. In this model, it's assumed
that the inertial mass density exhibits power-law dependence to its energy density
$$\rho_{inert} = \gamma\rho_0(\frac{\rho}{\rho_0})^{\lambda}$$
Where ##\gamma## and ##\lambda## are real...
The energy density of an EM wave is given as (1/2) ϵ E^2 + (1/(2μ)) B^2.
This is derived from the energy density of the electric and magnetic fields of capacitors and inductors, respectively.
But why should the energy density of the fields of capacitors and inductors be the same as that of...
Hello all,
Apologies in advance for the text-wall; this is a rather involved question.
I am trying to compute the effective transmission coefficient for a medium of non-uniform refractive index. For simplicity I am assuming the slab has thickness ##d##, that ##n(0)=1##, and that ##n(d)=n##...
So this is a question from my lab report on capacitance.
The aim of the experiment is to find out the relationship between surface charge density and radial distance from the centre of the plate capacitor. And in this experiment I have recorded 5 sets of data, namely r=0, V=4, r=1, V=3.5, r=2...
I have the solution for this problem but I did not understand the following statement:
The mass of water the crown displaced is ##m = 740- 690= 50 g##. Therefore the volume of the crown is ## 50 cm^3##
how can I conclude the volume of the crown from that displacement?
Hello, it's been a while since I've done any proper electrostatics, but I have a problem where I have a bunch of discrete point charges within some volume V bounded by a surface S.
I am wondering if it is possible to replace the discrete charge density in my volume V by some continuous surface...
I have the following probability density function (in Maple notation):
f (x) = (1 / ((3/2) * Pi)) * (sin (x)) ** 2 with support [0; 3 * Pi]
Now I want to transform x so that
0 -> (3/2) * Pi
and
3 * Pi -> (15/2) * Pi
and the new function is still a probability density function.
How should I...
For getting the density of states formula for photons, we simply multiply the density of states for atoms by 2 (due to two spins of photons). I am getting the 2D density of states formula as :- g(p)dp = 2πApdp/h^2
I think this is the formula for normal particles, and so for photons I need to...
Given the support [a, b] of a probability density function. How can I change the formula for the probability density function with a support [u, v]? Example: Given the beta distribution with support [a=0,b=1]:
$$\frac{x^{p-1} (1-x)^{q-1}}{Beta(p,q)}$$
Then the beta distribution with support...
My questions are as follows:
1. How do we find them and why do we need them?
2. What are the meanings of the mean and the median of a PDF? Are the formulae below correct?
$$\int_{a}^{median} f(x) \mathrm{d}x = \int_{median}^{b} f(x) \mathrm{d}x$$
$$\int_{a}^{mean} f(x) \cdot x \mathrm{d}x =...
Why is electric current not a vector while electric current density is a vector? What's the intrinsic difference between the two through that surface integral?
After computind dirac 1D equation time dependant for a free particle particle I get 2 matrixs. From both,them I extract:
1) the probablity matrix P =ps1 * ps1 + psi2 *psi2
2) the current matrix J = np.conj(psi1)*psi2+np.conj(psi2)*psi1
I think that current is related to electricity, and...
Hello everyone,
I have a math / physics question that has been with me for a while. I would be grateful if someone could help me.
Given a density matrix, what is the minimum value a sum of some of its off-diagonal elements can assume (or the most negative value)?
Remark: if one collect an...
I think the right choice is c. I'll pass on my reasoning to you:
We can think that if the formula of the potential is
V(r)=\dfrac{kq}{r}
If r tends to infinity, then V(r)=0.
But the correct answer is d).
My guess is that the force per volume is:
$$ \vec F_V = \rho \alpha x \hat x + \vec J \times \beta x \hat y$$
but I'm not sure where to go after that. I'm not given a value for either the charge density or the current density, so I can't simplify the relation much. Further, I'm not sure if my...
I have written a finite difference program to solve 1D time-independent Schrodinger equation. It seems to work correctly for harmonic oscillator, particle in a box, etc. But I can't figure out how to calculate the probability current density. It should be constant, but what is it? The program...
I do not have the solutions to this problem so I'm wondering if my attempt is correct.
My attempt at solution: We have two surfaces which we can calculate the area of. I think we can use gauss law to find the electric field and then integrate the E-field to find the electric potential.
So for...
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
$$E = \frac{h^2 \cdot...
I am trying to calculate the interaction energy of two interpenetrating spheres of uniform charge density. Here is my work:
First I want to calculate the electric potential of one sphere as following;
$$\Phi(\mathbf{r})=\frac{1}{4 \pi \epsilon_{0}} \int...
I am confused whether for electron I have to use rest mass energy (moc2 + 0.8 MeV) or just 0.8 Mev for calculating E.
Also how do I find minimum density of a neutron star using above data ? Please help !
I'm given the following density of states
$$ \Omega(E) = \delta(E) + N\delta(E-\Delta) + \theta(E-\Delta)\left(\frac{1}{\Delta}\right)\left(\frac{E}{N\Delta}\right)^N $$
where $ \Delta $ is a positive constant. From here I have to "calculate the canonical partition function as a function of $$...
The body is a small ball. The experiment consists in dropping this ball, while varying the diameter each 3 trials, in a viscous liquid and measuring its falling time: ##D_i\sim t_1,\,t_2,\,t_3##.
The equation we are using is:
$$\frac{\Delta x}{\Delta...