What is Charge distribution: Definition and 244 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.
Electrostatic potential $$ \Phi(\vec{r})=k \int \mathrm{d}^{3} r \frac{\rho\left(\vec{r}^{\prime}\right)}{\left\vec{r}\vec{r}^{\prime}\right} (i) $$ with $$ k=\frac{1}{4\pi\epsilon_{0}} $$ in SI units.
What work is required to move a point charge q from infinity to the center of the through...
I'm having trouble understanding how a charge distribution in a sphere can make this happen.
My instinct is that the fact that it's radially directed is a big hint of something, but I don't know what that hint might be alluding to. If the net Efield is constant inside the sphere and is always...
For 2D charge distribution ρ(x,y)=Ne PDF(x,y), where PDF is the normalized probability density function with its peak on (0,0) and has standard deviations σ x. and σ y. Are the contours with the equal probability "PDF(x,y)=const" the same as the equipotiential contours?, I tend to think that...
I know we're supposed to attempt a solution but I'm honestly super confused here. I think the second an third terms of the del equation can be cancelled out because there is only an E field in the r hat direction, so no e field in the theta and phi directions. That leaves us with ##\nabla \cdot...
How and why can charge be evenly or uniformly distributed in a conductor? How can such near perfect configuration of charge be achieved? Is outside influence (or force) or any special scientific tools or instruments required to accomplish that? By definition, electrostatic equilibrium is...
I understand part (a) of this question, and my answer for that part is:
*For r < a*
E = (ρ0 * r4) / (6 * ε0 * a3)
* For r ≥ a*
E = (ρ0 * a3) / (6 * ε0 * r2)
Now, for part (b), I understand one solution is, for r < a, find the work done to bring a point charge q from infinity to a and then from...
Hello!
I am trying to solve this exercise of the electric field, but it comes out changed sign and I don't know why.
Statement: On a straight line of length ##L=60\, \textrm{cm}## a charge ##Q=3,0\, \mu \textrm{C}## is uniformly distributed. Calculate the force this linear distribution makes...
In this question it is given that the sphere which is conducting is initially given a charge q then due to nonuniform mechanical strength and due to electrostatic force it creates a Small hemispherical bulge on its surface?
okay my doubt is Let me define a term σ where σ is surface density...
Gauss' law: $$\iint_{\partial A}\vec E\cdot d\vec A=\frac{Q}{\epsilon_0}$$
Suppose we have a unevenly charged nonconducting spherical shell, in which a Gaussian surface is placed. In this case, is the electrical field on A 0, given that there is no charge inside A? I came up with this example...
Hello! I am a junior undergraduate physics major and I am very confused on how to visualize things in my electrodynamics class. Specifically, I am having issues with dielectrics and spheres with constant potentials etc. I usually notice that I am lost in a class when I can no longer draw out a...
All I can say is that where the charge density on surface is higher, we will have a stronger electric field compared to areas where charge density is lower since more charges means greater electrical force on a test charge placed very close to the surface.
Also, the potential on pointed areas...
This doubt is confusing to me.
I know it's something to do with conductors and insulators, but cannot explain. Conductors have mobile/free electrons unlike insulators. Having free electrons doesn't seem to explain this difference of charge distributions.
A spherical volume charge (R<=1cm) with uniform density ρv0 is surrounded by a spherical surface charge ( R=2cm) with charge density 4 C/m2. If the electric field intensity at R=4cm is 5/Є0 ,deterime ρv0
According to my professor, the solution in this book (pages 2021) for item (ii) is wrong: https://www.ucursos.cl/usuario/75468645ed16a71af6da3ffd813d47f5/mi_blog/r/Problems_and_Solutions_on_Electromagnetism.pdf
a) Static charge distribution should result in a static electric field? Legitimacy should be checked with curl of E = 0?
b) Using the second equation should give is the answer?
Hello everyone,
I am new to this site so I hope this is the right place to ask this. I understand simulating electric field intensity using electrostatics because E=V/d makes sense to me. I do not understand how to consider efield intensity using charge distribution. When is charge...
I sort of understand the meaning of this integral, but I don't know how to evaluate it. I have never evaluated a volume integral. It would be very helpful if someone could explain in other words what this integral means and give an example evaluating it.
This is from Purcell's Electricity and...
I quite understand the fact the EPE (Electrical Potential Energy) of a system of two charges are U = k*qQ/r, Q is fix. however when it comes to three charges i get lost. because my reasoning is :
if q1 is fix then the EPE of the system when q2 is brought is U2 = k*q1*q2/r12, when q3 is brought...
I'm just going to skip some of the step since I only need help with understanding the last part.
After rearranging the equation stated at "Relevant equation" (and skipping some steps) we will get:
E * 4*pi*e0*R^2 = integral pv * 4*pi*R^2 dR
E = 1/(4*pi*e0*R^2) * 4*pi * integral pv*R^2 dR
E =...
My answer was +Q/3.
I was assuming that the charges would distributed themselves completely.
But, apparently, I'm wrong?
For example, if there were 12##e^##s on Sphere C, then, in the first step in the system: the ##e^##s would balance out until each sphere has 4 ##e^##s each?
What am I...
This is not really homework, but I'm having trouble understanding it intuitively. I came across this when learning about the space charge layer of a diode. The solution I know simply uses the 1D form of Gauss's law: ##\vec{\nabla} \cdot \vec{E}## = ##\dfrac{\rho}{\epsilon_0}## becomes...
Hello, I have a problem where I'm supposed to calculate the charge distribution ρ. I need to calculate it by applying the Laplacian operator to the potential Θ. The potential is the function: q*exp(αr)/r
I found on the internet that for this type of potentials I cannot just apply the...
I was trying to calculate the EMFs from power lines, just to see how they correspond to transmission line right of ways, and got a little stuck calculating the electrostatic Efield (∇V) from power lines. I know it is dependent on the charge distribution on the power line, which is in turn...
I'm working through Griffiths EM 3rd ed. in section 2.4.2 (point charge distribution) and 2.4.3 (continuous charge distribution).
I understand from the section on point charge distributions that when we add up all the work (excluding the work necessary in creating the charge itself), one clever...
Does anybody know if there is an analytical expression for the electrostatic potential produced by a charge distribution confined to a double cone shaped region. Think of a beam of charged particles converging to a focus and then diverging again. The total charge in each thin, crosssectional...
Consider a very simple idealized circuit, with a constant voltage emf, perfectly conducting wires and a resistor all in series. There is a potential drop across the resistor, given by Ohm's law: ##V = IR##. I have read on the Internet that many people say that the potential drop is caused by a...
Two identical metalic spherical conductor of radii ##R## are at a distance ##d## apart.One of the conductor has charge ##Q## while the another one is neutral.What will be the induced charge on the other conductor ?
If we put an image charge ##q## inside the neutral one. Then the potential at...
Dear colleagues
I have this problem which I don't understand from where they got the solution I tried to solve it with slot of methods with the same answer which not the stated answer.
A point charge (q) is located a distance (b) from a grounded conducting sphere with radius (a) show that the...
Homework Statement
I need help on solving this exercise :
We have a ring of radius = ##a## uniformly charged (total charge = ##Q##) and on its axis a segment ##OA## (length = ##a## also) of uniformly distributed positive electric charges with the charge density ##\lambda'## and of total charge...
Homework Statement
Homework Equations
dV= integral(kdQ/dR)
The Attempt at a Solution
So, I'm familiar with these type of problems but in 2D (like a line of uniform charge).
When the y,z component is added, I'm kinda lost.
i know dQ = p*dV= p*dx*dy*dz. (atleast i think it is).
also the dR =...
Homework Statement
Working through Purcell (among others) as fun applied math/math modeling refresher. But, I have struggled all week in establishing from first principles that the potential/field/distribution for a configuration of two capacitive disks of radius 1 and separation s along the...
Homework Statement
Two thin conducting spherical shells have radii R1 and R2.Outer shell is charged to q and inner shell earthed.Find charge appearing on both the shells.
Homework Equations
The Attempt at a Solution
Isnt the charge on inner shell 0 and charge on outer shell remains Q as it...
Homework Statement
[/B]
For a straight wire of length 2L carrying a uniform charge density ##\lambda##, find
1) potential a distance z above the centre
2) electric field E at that point,
3) energy of this charge distribution
Homework EquationsThe Attempt at a Solution
1) and 2) I can do...
Homework Statement
Find the distribution of charge giving rise to an electric field whose potential is $$\Phi (x,y) = 2~tan^{1}(\frac{1+x}{y}) + 2~tan^{1}(\frac{1x}{y})$$where x and y are Cartesian coordinates. Such a distribution is called a twodimensional one since it does not depend on...
I have a simple AC circuit. For example a battery with a capacitor.
In the steadystate the capacitor has the same V of the battery.
The wire that connect the two components is a very small cylindrical conductor, so it should have a surface charge distribution. It's that correct ? If no, why ?
Homework Statement
part b of below
[/B]
Homework Equations
##(1+x)^{1/2}=1+\frac{1}{2}x\frac{x^{2}}{8}+...##
The Attempt at a Solution
[/B]
##\int\limits^{\Lambda}_{\Lambda} \frac{dy}{\sqrt{r^2+y^2}}=log(\lambda+\sqrt{\lambda^2+r^2})  log(\lambda+\sqrt{\lambda^2+r^2}) ##
##=...
Is the potential energy of a symmetric planar (x,y) charge distribution lower than any non symmetric distribution ? from the discussion on Gauss's law and symmetric charge distributions I would think so because the electric field could only be normal to the (x,y) plane in the symmetry case but...
Greetings! I'm new here and I think about this place as soon as I see what the statement asks.
Homework Statement
Considering the volumetric density ρv=(e2r/r2), figure the total charge (ℚ) of the universe.
Homework Equations
[/B]
ρv=ΔQ/ΔV > (ΔQ ∝ ΔV)
ℚ=∫v ρv dxdydz
The Attempt at a...
Homework Statement
Two conducting spheres having same charge density and with radius “R” & “2R” are brought in contact and separated by large distance. What are their final surface charge densities ?
Homework Equations
No equation in question.
The Attempt at a Solution
Tried using the fact...
1. The problem statement, all variables, and given/known data
Suppose you have a wire of length l and a uniform line charge density λ. Find the electric field at the midpoint that is height r above the xaxis
Homework Equations
(see attached)
The Attempt at a Solution
To solve, I used the...
Homework Statement
Hello,
this is more of a conceptual question than a concrete homework assignment question. I'm learning about Gauss's law and the Prof did an exercise on a sphere with uniform charge distribution, where he found E(r). The trick was, that E(r) was constant over the Gaussian...
Homework Statement
In a certain region, the electric potential due to a charge distribution
is given by the equation V (x, y, z) = (3x2y2+yz32z3x)V0/a4 where
a, x, y, and z are measured in meters and V and V0 are in volts. What
is the magnitude of the electric field at the position (x, y, z)...
Homework Statement
A metallic sphere of radius a is placed concentrically with a metallic spherical shell with inner radius b and outer radius c. The sphere has a total charge of 2Q and the shell has a total charge of 3Q.
(a) What is the charge distribution? Specifically, what is...
Homework Statement
Initially there is a spherical charge distribution of with a radius ##R_0## and uniform charge density ##ρ_0##. Suppose the distribution expands spherically symmetrically such that its radius at time t is ##R_0 + V t##, where V is the velocity. Assuming the density remain...
Hey all,
So the question in Jackson 1.4 is that I have 3 spheres that all have a total charge Q on them, but each sphere has different material properties. For instance, I have a conducting sphere, a sphere with a uniform charge distribution, and one with a charge distribution that has a...
Homework Statement
[/B]
Volume charge density in some space is given by a function ##ρ_v(x)=ρ_0\frac{x}{a}e^{\frac{x^2}{a^2}}## where ##ρ_0, a## are positive constants. Determine the electric field vector in arbitrarily chosen point in space.
Homework Equations
3. The Attempt at a Solution...
Homework Statement
Hi everybody! I'm very stuck trying to solve this problem, hopefully some of you can give me a clue about in which direction I should go:
Determine the multipole expansion in two dimensions of the potential of a localized charge distribution ##\lambda(\vec{x})## until the...
This is my first time using this site so please excuse me if I missed any guidelines.
1. Homework Statement
A plastic rod having a uniformly distributed charge Q=25.6pC has been bent into a circular arc of radius R=3.71cm and central angle ∅=120°. With V=0 at infinity, what is the electric...
Homework Statement
Consider the charge distribution of a uniformly charged ring of radius ##R## and charge ##Q## at a distance ##d## above the origin and a uniformly charged ring of radius ##R## and charge ##Q## at a distance ##d## below the origin.
(a) Calculate the dipole moment of this...