What is Charge density: Definition and 439 Discussions
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m−3), at any point in a volume. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m−2), at any point on a surface charge distribution on a two dimensional surface. Linear charge density (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m−1), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative.
Like mass density, charge density can vary with position. In classical electromagnetic theory charge density is idealized as a continuous scalar function of position
x
{\displaystyle {\boldsymbol {x}}}
, like a fluid, and
ρ
(
x
)
{\displaystyle \rho ({\boldsymbol {x}})}
,
σ
(
x
)
{\displaystyle \sigma ({\boldsymbol {x}})}
, and
λ
(
x
)
{\displaystyle \lambda ({\boldsymbol {x}})}
are usually regarded as continuous charge distributions, even though all real charge distributions are made up of discrete charged particles. Due to the conservation of electric charge, the charge density in any volume can only change if an electric current of charge flows into or out of the volume. This is expressed by a continuity equation which links the rate of change of charge density
ρ
(
x
)
{\displaystyle \rho ({\boldsymbol {x}})}
and the current density
J
(
x
)
{\displaystyle {\boldsymbol {J}}({\boldsymbol {x}})}
.
Since all charge is carried by subatomic particles, which can be idealized as points, the concept of a continuous charge distribution is an approximation, which becomes inaccurate at small length scales. A charge distribution is ultimately composed of individual charged particles separated by regions containing no charge. For example, the charge in an electrically charged metal object is made up of conduction electrons moving randomly in the metal's crystal lattice. Static electricity is caused by surface charges consisting of ions on the surface of objects, and the space charge in a vacuum tube is composed of a cloud of free electrons moving randomly in space. The charge carrier density in a conductor is equal to the number of mobile charge carriers (electrons, ions, etc.) per unit volume. The charge density at any point is equal to the charge carrier density multiplied by the elementary charge on the particles. However, because the elementary charge on an electron is so small (1.6⋅10−19 C) and there are so many of them in a macroscopic volume (there are about 1022 conduction electrons in a cubic centimeter of copper) the continuous approximation is very accurate when applied to macroscopic volumes, and even microscopic volumes above the nanometer level.
At atomic scales, due to the uncertainty principle of quantum mechanics, a charged particle does not have a precise position but is represented by a probability distribution, so the charge of an individual particle is not concentrated at a point but is 'smeared out' in space and acts like a true continuous charge distribution. This is the meaning of 'charge distribution' and 'charge density' used in chemistry and chemical bonding. An electron is represented by a wavefunction
ψ
(
x
)
{\displaystyle \psi ({\boldsymbol {x}})}
whose square is proportional to the probability of finding the electron at any point
x
{\displaystyle {\boldsymbol {x}}}
in space, so

ψ
(
x
)

2
{\displaystyle \psi ({\boldsymbol {x}})^{2}}
is proportional to the charge density of the electron at any point. In atoms and molecules the charge of the electrons is distributed in clouds called orbitals which surround the atom or molecule, and are responsible for chemical bonds.
We know Gauss's law for an infinite sheet yields ##\textbf{E}=\frac{\sigma}{2\varepsilon_{0}}##. This is relatively elementary and I completely understand the derivation. This is also valid when looking at a parallel plate capacitor (the electric field is additive between the plates yielding...
This is SAMPLE PROBLEM 257 from "Physics" by Resnik, Halliday, and Krane, in the chapter "Electric Field and Coulomb's Law".
After describing the behavior of uniformly charged spherical shells:
follows a sample problem:
The solution to (a) goes to say that the volume inside R/2 is 1/8 of the...
Here's my attempt at a solution, but when I plug it in, it gives me a power ten error. I don't really understand what I'm doing wrong here. I think all my variables are in the correct units and it asks for my answer to be in μC/m2. Any help is much appreciated.
Okay so I am a little confused as to where I made a mistake. I couldn't figure out how to program Latex into this website but I attached a file with the work I did and an explanation of my thought process along the way.
Schrodinger’s original interpretation of the wavefunction was that it represented a smeared out charge density however this was replaced with Max Born’s probability interpretation. The issue was from what I understand that a charge density would repel and have self interactions as all the charge...
For part a:
I know that linear charge density is the amount of charge per unit length, and we are given the volume charge density. Since we are given the volume, we can obtain the length by multiplying the volume by the cross sectional area, so C/m^3 * m^2 = C/m. The cross sectional area of a...
Part (a) was simple, after applying
$$Q=\int_{\mathbb{R}^3}^{}\rho \, d^3\mathbf{r}$$
I found that the total charge of the configuration was zero.
Part (b) is where the difficulties arise for me. I applied
$$V(\mathbf{r})=\frac{1}{4\pi \epsilon _0}\int_{\Gamma }^{}\frac{\rho...
I've attached what I have so far. Used Gauss's law, everything seemed to make sense except the units don't work out in the end. The charge density function if given by: r(z)=az, where z is the perpendicular distance inside the plane.
According to Helmholtz’s theorem, if electric charge density goes to to zero as r goes to infinity faster than 1/r^2 I'm able to construct an electrostatic potential function using the usual integral over the source, yet I don't understand how this applies to a chunk of charge in some region of...
I have tried to solve the problem by setting as a condition that the electric field inside the conductor has to be 0, but in this way I have two unknowns (σ1 and σ2):
Hi! I've been trying to attempt this problem over here but the solutions state that the solution is this below?
However, from integrating the density and then plugging it into Gauss's law, I get the exact same thing, except a 15 instead of a 5. Could any please help point out if there is an...
$$\phi_E=\dfrac{Q_{\textrm{enclosed}}}{\varepsilon_0}\Rightarrow Q_{\textrm{enclosed}}=9,6\cdot 10^{7}\, \textrm{C}$$
$$Q_{\textrm{enclosed}}=\sigma S=\sigma \pi R^2\Rightarrow \sigma =\dfrac{Q_{\textrm{enclosed}}}{\pi (0,1^2)}=3,04\cdot 10^{5}\, \textrm{C}/\textrm{m}^2$$
I have a lot of...
Hey, I have a really short question about electrostatics.
The boundary conditions are :
\mathbf{E}^{\perp }_{above}  \mathbf{E}^{\perp}_{below} = \frac{\sigma}{\varepsilon_{0}}\mathbf{\hat{n}} ,
\mathbf{E}^{\parallel }_{above} = \mathbf{E}^{\parallel}_{below}.
My question is what is...
Hi,
I have a dialectric cube and inside the center of the cube I have a part where we have Introduced evenly electrons.
I have to find the polarization charge density in the 3 regions.
I know outside the cube is the vacuum, thus ##\vec{P} = 0## and inside the dialectric (non charged part)...
There are some question involving the statement. One of them is about the charge density in S' frame. It asks to calc it.
I thought that i could calculate the electric field in the referencial frame S' and, then, use the formula
$$ E = \lambda / 2 \pi \epsilon l $$
In that way, i would obtain...
hi guys
our professor asked us to confirm the units of volume charge density ρ and also the surface charge density σ of a dielectric material given by
$$
\rho = \frac{1}{4\pi k} \vec{E}\cdot\;grad(k)
$$
$$
\sigma= \frac{(k1)}{4\pi} \vec{E_{1}}\cdot\;\vec{n}
$$
I am somehow confused about the...
It is given that the charge density of a particle of charge ##q_0##, world line ##z^{\mu}(\tau)## (and 4velocity ##u^{\mu}##) in a spin##s## force field is a ##s##tensor\begin{align*}
T^{\mu \nu \dots \rho}(x^{\sigma}) = q_0 \int u^{\mu} u^{\nu} \dots u^{\rho} \delta^4[x^{\sigma} ...
I have already calculated the polarisation that is
$$ \mathbf{P} = \frac{\rho_f r}{2} \left( 1  \frac{\epsilon_0}{\epsilon} \right) \hat{r} . $$
I tried to use the following formulas to calculate the density bound charges. For the surface bound charge I got:
$$ \sigma_{b1} = \mathbf{P} \cdot...
Guys I have Problems with this task The arrangement consists of a point charge Q at a distance (x0, y0,0) from the origin and two perfectly conductive surfaces in the (x, z) and (y, z) plane
a) Mathematical description of the space charge density p of the original and mirror charge using the...
Okay, so I tried thinking of this as like a simple balancing of equations. There's an infinite sheet of charge on the left and a conductor on the right with some charge already on it. My thought process was that the side nearer to the charged sheet would have 4.7 more μC/m2 than the far side...
In this page you can see it’s written: E must be perpendicular to the surface. If it were not then the charges would move due to a component along the surface.
I am assuming the field is generated due to the charges on the surface.
I have a doubt. Is the author saying if it were not...
Hello, I was reviewing a part related to electromagnetism in which the charge and current densities are defined by the Dirac delta:
##\rho(\underline{x}, t)=\sum_n e_n \delta^3(\underline{x}  \underline{x}_n(t))##
##\underline{J}(\underline{x}, t)=\sum_n e_n \delta^3(\underline{x} ...
First I wrote in ##S'##, by using Gauss theorem
$$
\int_{\Sigma} \underline E' \cdot \hat n d\Sigma = \frac Q {\varepsilon_0} \rightarrow E'(r)2\pi rH=\frac{\lambda'H}{\varepsilon_0}
$$
$$
\underline E'(\underline r)=\frac{\lambda'}{2\pi\varepsilon_0r}\hat r
$$
Its components are...
Hey guys! Sorry if this is a stupid question but I'm having some trouble to express this charge distribution as dirac delta functions.
I know that the charge distribution of a circular disc in the ##xy##plane with radius ##a## and charge ##q## is given by $$\rho(r,\theta)=qC_a...
There is a section in the BJT explanation the charge density and the corresponding electric field graphs. But i was not sure how the electric field is derived and hence i started deriving it. Please correct me if my understanding is wrong in posting the question
It is an ##npn## BJT. My...
In this explanation we need to involve the Dirac delta functions(maybe) but I clearly have a difficulty in understanding it can some one explain me the whole concept of constant or non constant volume charge density.
The answer is that the charge density would be σ, I cannot for the life of me understand why would that be the case. Of course it makes sense but I can't convince myself that it would be the only possible answer.
I have tried to apply Gauss law a few times, but it doesn't yield anything.
Electric field for the semicircle
$$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}})$$
In this case E is equals to 8 N/C
What I'm searching is R, λ, and the length of the wire, so I think...
I'm not sure I understand why I need to use ##d##.. Maybe they want me to have the potential be zero at ##A##?
In any case, I have found$$V(B)=\alpha k\int_0^L\frac{x}{\sqrt{b^2+\left(x\frac{L}{2}\right)^2}}dx+C=\frac{\alpha...
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...
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 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 Efield to find the electric potential.
So for...
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;
$$\Phi(\mathbf{r})=\frac{1}{4 \pi \epsilon_{0}} \int...
A rod with a circular center in the middle (which causes the rod to change direction by 90 °) has an evenly distributed linear charge density 𝜆 of electrons along the entire rod. Determine the electrical potential of the red dot in the figure below which is at the center of the circular round...
Based on the conditions, I found that $$V(x)=\frac{a^2}{\pi^2} ρ_0sin(πx/a)$$ would be a solution to Laplace's equation for $$x\leq a$$
and $$V(x)=cx+d$$, where c and d are constants. From the boundary conditions, $$\frac{dV(a)}{dx}=\frac{a}{\pi} ρ_0cos(πa/a)=ac$$, $$c=\frac{a\rho}{\pi}$$ and...
In most standard exposition of (the meanfield theory of) charge density wave (CDW), phase and amplitude fluctuations are introduced as the collective excitations. Kohn anomaly in the acoustic phonon dispersion is also mentioned as temperature goes from the above till the CDW transition...
The textbook says
' A conducting sphere shell with radius R is charged until the magnitude of the electric field just outside its surface is E. Then the surface charge density is σ = ϵ0 * E. '
The textbook does show why. Can anybody explain for me?
Here is my work done for this problem, along with a diagram of the situation. I'm not worried so much about the arithmetic because our tests are only 50 min long so the problems they give us do not require heavy integration or calculus, but you need to know what goes where in the formula. That...
E0=V/d = 100/0.1 =1000v/m
In slab 1, E1=E0/k1=500v/m
In slab 2, E2=E0/k2=250v/m
Applying Gauss' Law to a box surface surrounding the interface with area equal to the plates we have
(E1+E2)*A = Q/epsilon_naught
So charge density sigma = 250 epsilon_naught
But answer given is...
Homework Statement
The charge of uniform density 50 nC/m3 is distributed throughout the inside of a long nonconducting
cylindrical rod (radius = 5.0 cm). Determine the magnitude of the potential difference of point A (2.0 cm from the axis of the rod) and point B (4.0 cm from the axis).
a . 2.7...
Homework Statement
A charge Q is uniformly distributed along the xaxis from x = a to x = b. If Q = 45 nC,
a = –3.0 m, and b = 2.0 m, what is the electric potential (relative to zero at infinity) at the point, x = 8.0 m, on the x axis?
a . 71 V
b. 60 V
c. 49 V < correct answer
d. 82 V...
Hi everyone, this is my first thread!
I am currently undergoing a personal investigation that is based on one of the factors which effect the splitting of d orbitals in central metal ion by the charge density of ligands (in a complex ion).
However, recently I got stumped by trying to...
Homework Statement
Consider a charged body of finite size, (\rho=0 outside a bounded region V). \vec{E} is the electric field produced by the body. Suppose \vec{E} \rightarrow 0 at infinity. Show that the total selfforce is zero: \int_V \rho \vec{E} dV = \vec{0}, i.e. the charged body does not...
Homework Statement
Homework Equations
While solving this problem at r >>a ,the corresponding potential due to the dipole is kpcosθ/r2(potential due a dipole) where k is the electrostatic constt. ...(1)
If σ(θ) is the surface charge density induced due to external electric field.
then the...
Homework Statement
[/B]Homework Equations [/B]
∫Dperpendiculards=qenclosed freecharge
D=ε0E+P
The Attempt at a Solution
D1+D2=q/2πr2
at a distance r from the centre
How to find D2 which is at the lower boundary e.g inside the dielectric??
Homework Statement
A charge Q is uniformly distributed throughout a nonconducting sphere of radius R. Write the expression of the charge density in the sphere?
Homework Equations
Charge density ρ=dQ/dV
Gauss's Law ∫EdA = E(4ϖr^2)
The Attempt at a Solution
If Q is uniform then ρ=Q/dV and the...
Homework Statement
Sphere 1 has net positive charge Sphere 2 has net negative charge Sphere 3 has net positive charge
The ranking of net charge magnitudes are
SPHERE 3 > SPHERE 2 > SPHERE 1
All spheres are conductors
Sphere 2 is moved away from Sphere 1 and toward Sphere 3 so that 2 and 3...