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.
Surface charge density is denoted with a lower case sigma: σ
Though sometimes I also see it denoted as σ with a subscript 0.
What is the difference in these notations?
Thank you.
Homework Statement
A line of charge starts at x = +x0 and extends to positive infinity. Consider two situations: tal electric flux through the paraboloidal L
(a) a uniform linear charge density λ = λ0, uniform electric field of magnitude E 0 in
(b) a coordinate-dependent linear...
Homework Statement
Using the method of images, solve the following
The Attempt at a Solution
I have very little idea on how to solve this. I looked up various methods online and none seem to be what the lecturer has used.
This is the solution he posted up, but as usual it is missing 300...
Homework Statement
"as long as the volume charge density is finite (which is not true of surface charge distributions or point charges), the electric field is continuous.
Homework Equations
The Attempt at a Solution
I know that for surface charges distributions and point charges...
Hello,
I am confused about how charge density is only related to the material of a conductor and not on any external factors.
I don't see how the molecular weight, density, and avogadro's number dictate the charge density.
Any help is appreciated, thanks
Homework Statement
a)Calculate the total charge a square capacitor plate would have with width x, height y, thickness z, and charge density f(x,y,z) = 1+x+y
b)Calculate the total charge a sphere would have with radius r, and charge density f(x,y,z)=x+y+z
Use the triple integration seen in...
We know that ∇.D = ρV
My question is, if D is due to a linear or a surface charge distribution, can we apply this equation and get a surface or linear charge density?
Thank you very much
Gauss' Law problem! Help please! infinite sheet with charge density?
In the figure below, a small circular hole of radius R = 1.80 cm has been cut in the middle of an infinite, flat, nonconducting surface that has uniform charge density σ = 4.50 pC/m2. A z-axis, with its origin at the hole's...
help with practice test problem! curved rod with charge density??
Two plastic rods are curved. The left rod is a 120° circular arc of radius
R, and has uniform charge density λ. The rod on the right is a 120° circular
arc of radius 2R, and has uniform charge density -2λ. A point P is located...
Let's say you have a sphere which has a charge distribution where the charge behind a radius r can be expressed as Q(r). You also have the volume formula for a sphere, V(r).
Why is ρ, the charge density, defined as: ρ=dQ(r)/dV(r) instead of simply ρ=Q(r)/V(r)?
Homework Statement
A line charge starts at x = +x0 and extends to positive infinity. The linear charge density varies inversely with distance from the origin, λ(x)=(λ0*x0)/x
derive the expression for the electric field at the origin, E0, due to this infinetly long line-charge (L→+∞)...
Homework Statement
Given a sphere of radius R with a volumic charge distribution p find the function p.
The electric field inside of the sphere is radial with a constant modulus E
Homework Equations
div E = p / ε
∫∫E.ds = 1/ε * ∫∫∫ p * dV
The Attempt at a Solution
I tried to solve it...
hey guys and gals, I'm normally pretty good at my physics homework, but this problem has my STUMPED, I think I am starting it right, but I have no idea how to proceed or if I am on a good path.
Homework Statement
A Very Long Cylinder with a radius of 7.2cm (.072m) has a uniform density of...
Homework Statement
A proton orbits a long charged wire, making 1.30*10^6 revolutions per second. The radius of the orbit is 1.20cm.
What is the wire's linear charge density?
Homework Equations
- q E = m w^2 r
- 9*10^9 [2 λ /r] q = m w^2 r
The Attempt at a Solution
λ = linear...
1. A spherically symmetric charge distribution results in an electric potential of the form
What is the charge distribution?
2.
Hint: consider the difference in electric field between two values of r
Show that the answer is of the form
3. I have attempted several solutions but haven't...
Homework Statement
Consider a spherical shell with radius R and surface charge density σ = σ0 cosθ
(a) What is the total charge carried by the shell?
(b) Please evaluate the charge carried by the upper hemisphere, in terms of σ0.
Homework Equations
Q=∫σ0 cosθ da
The...
I feel that I'm always struggling with these kinds of problems since I never know if the information is to be considered a density or not. Is there any trick/key words I should be on the look-out to determine whether they gave me a density charge or a point charge?
Homework Statement
The sphere has radius R, and uniform volume charge density P. This sphere remains stationary (levitates) when placed above an infinite sheet of paper with a uniform surface charge density u. What is this sphere's mass? Homework Equations
4/3 pi R^3 is the volume of a sphere...
I have a conceptual problem.
Homework Statement
I was given a charge distribution for an electron cloud of a hydrogen atom in the ground state - ignoring the nucleus.
Homework Equations
charge density: ρ=charge of electron/(pi*Bohr radius^3)*exp(-2r/Bohr radius)
The Attempt at a...
Homework Statement
A solid ball of radius r1 has a uniform charge density ρ.
a) What is the magnitude of the electric field E(r) at a distance r>r1 from the center of the ball?
b) What is the magnitude of the electric field E(r) at a distance r<r1 from the center of the ball?
Express your...
Homework Statement
The problem statement is given in the figure below
Homework Equations
E=σ/(2* ε)
The Attempt at a Solution
The top plate is assumed to be negative charge. The bottom plate is assumed to be positive charge. Since negative charge pulls and positive charge pushes...
I'm interested in working out an efficient and precise numerical method to find the charge density on a conductor with an axially symmetric shape that is essentially a deformed sphere. (I have in mind something like r(\theta)=\sum \beta_\ell P_\ell(\cos\theta), where P is a Legendre polynomial.)...
Homework Statement Find the electric charge centred in a sphere of radius a, centered at the origin where the electric potential is found to be (in spherical coordinates) V(r)=kr^-2 where k is some constant.
The Attempt at a Solution
We have E=-\nabla V = -2kr^{-3} \hat{r}
So...
Homework Statement
Solid ball of charge with radius R and volume charge density ρ(r) = ρ0r2, centred at the origin.
I have already found the electric field for r<R and r>R and also the potential at the origin by using the formula:
V = -∫E.dl
Now i want to find the potential at the...
We have all seen Ohms law, J=σE. This approximations makes sense in simple electric fields in which the charges are accelerated in parallel.
However as I will demonstrate, this implies a few conditions on the charge density (ρ) associated with the current density (J).
Now, from the continuity...
Homework Statement
A line charge in the x directio has a variable charge density given by the equation λ=4 λnot((1-x)/2L), where λnot is a constant. The rod has a length of L.
i)What is the net charge of the rod? Hint-the net charge is calculated by integrating the charge density with respect...
Homework Statement
Show that electrostatic field E = br
where b is a constant
is irrotational
Find the charge density ρ(r) which can generate this electric field
Homework Equations
∇ x E = 0 (following stoke theorem)
ρ = ε0(div E) since: div E = \frac{ρ}{ε}
The Attempt...
So the problem is: The electrostatic field is E =g|r|r where g is a constant. Find the charge density p(r) which can generate the field?
My attempt was gauss law: divE=p/ε0
But I'm struggling to take divE. I guess r=x+y+z?
Any advice much appreciated!
As shown in the figure, a rod of length 9.8 m lies along the x-axis, with its left end at the origin. The rod has a non-uniform linear charge density λ = αx, where α = 0.009 C/m2 and x is the position. Point A lies on the x-axis a distance 3.59 m to the left of the rod, as shown in the figure...
The divergence of electric field at a point is proportional to the charge density at the point. Divergence is the rate of change with distance, the rate of change of electric field due to a distant charge is not zero, so how can it be said that the divergence at a point depends only on the...
A NONCONDUCTING sphere with uniform volume charge density ? HELP!
Homework Statement
so this is the problem,
a nonconducting sphere of radius \alpha has a uniform volume charge density with total charge Q. the sphere rotates about an axis through its center with constant angular velocity...
Homework Statement
From the picture attached it can be seen that there are two plates with charge density σa and -σb. We will chose → as the n direction.
What is that electric field everywhere for this system?
Homework Equations
∫∫E.dS = ∫∫σ/ε0 dA
The Attempt at a Solution
Using Gauss' law...
I have no clue how to start this problem. The professor wrote:
"Write the charge density of a ring."
...and that's it.
I know it would probably be ρ as a function of the radius. But I don't know how to move forward. I was looking through the early section in Griffith's E&M Ch. 3...
As you probably can see from the above shot, I'm determining charge density via the Laplacian over the potential (phi). I understand the mathematical steps, just confused on the factor of 4pi that pops up in the denominator. I think I understand why you would do that and here's my reasoning...
Homework Statement
Consider charge distribution \rho = \frac{A}{r} with spherical symmetry, for 0 \leq r \leq R, and \rho = 0 for r > R, and A is a constant. Find the Electric Field in all of space. Check your answer obtaining \rho from your answer.
Homework Equations
Gauss's law...
Firstly, I have been able to find almost nothing on this kind of question in textbooks or online anywhere. Most places (including my lecture notes) give at most the definition of the operator and that's all. One page if you're lucky out of a whole book. I'd kill for some examples, if you could...
So I bugged the folks in General Physics about the latter form of the question a while back, and got some rather unconvincing "can't be done" replies. To state the problem specifically (and my motivation):
Let's say that the probability density of finding a particle at any place/time is given...
Homework Statement
A nonconducting wall carries charge with a uniform density of 8.55 µC/cm2.
(a) What is the electric field 8.55 cm in front of the wall if 8.55 cm is small compared with the dimensions of the wall?
Homework Equations
σ= Q/A
E=σ/2Eo
The Attempt at a Solution...
In statistical mechanics, a probability current (http://en.wikipedia.org/wiki/Probability_current) can be derived from the probability amplitude via
ψ*ψ,i - ψψ*,i (i=x,y,z)
...and it would be nice or any number of reasons to be able to do the same thing with a non-probability...
Problem Statement: I'm having some trouble understanding how to write charge densities using delta functions, particularly when they involve geometries other than Cartesian. So I have a disk moving with velocity v (along the z-axis) that has total charge Q, and I'm trying to write ρ(x,t) so that...
In the diagram attached, the 3 resonance structures of arenium ion bonded to H electrophile are shown.I labelled the carbon atoms in one of the structures for convenience.
From the diagram, the A and B carbons, and D and E carbons, have a double bond between them for two of the resonance...
This is for extra credit not actual homework but I need the points and I would like to understand the subject matter as well so any help would be much obliged.
Homework Statement
There is a sphere with radius R made up of a perfect conducting material in a constant and uniform electric field...
Homework Statement
This is not really a homework just studying but I'm kinda stuck.
So I am trying to find out how to formally write down the Charge Density for any distribution.
Although I will not get into Green's Function or how to find V, I got that fine.
My example will be a Rod of...
Homework Statement
Two large parallel metal plate sheets carrying opposite electric charges of equal magnitude are separated by a distance of 38.0mm. The electric field between them is uniform and has magnitude 480 N/C.
a) What is the potential difference between the sheets?
b) Which sheet is...
Say you are given the problem of constructing the charge density vs position graph of a BJT(lets take a NPN BJT under equilibrium conditions for example). Now, it is a given as to how a NPN BJT Energy Band Diagram would look like through intuition or a google search. Using only the information...
Hey guys :)
How can I calculate the charge density with the PDE Toolbox, when I know the overall charge or the electrical potential of the areas? I want to know how the charges in two conducting spheres align when they are near to each other. I'm looking for a simple graphical way to play...
Homework Statement
There is a region of space containing a uniform electric field of strength E. A neutral conducting cube with side length s is placed into this field. The cube is aligned with the field. You may assume that the electric field outside the cube is unaffected by any changes that...
Homework Statement
A long, non conducting, solid cylinder of radius 4.2 cm has a nonuniform volume charge density ρ = Ar^2, a function of the radial distance r from the cylinder axis. A = 2.5 µC/m5.
(a) What is the magnitude of the electric field at a radial distance of 3.2 cm from the axis...