What is Surface charge density: Definition and 86 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


{\displaystyle {\boldsymbol {x}}}
, like a fluid, and




{\displaystyle \rho ({\boldsymbol {x}})}




{\displaystyle \sigma ({\boldsymbol {x}})}
, and




{\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




{\displaystyle \rho ({\boldsymbol {x}})}
and the current density





{\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




{\displaystyle \psi ({\boldsymbol {x}})}
whose square is proportional to the probability of finding the electron at any point


{\displaystyle {\boldsymbol {x}}}
in space, so







{\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.

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  1. Wmdajt

    E&M: Gauss' Law Surface Charge Density

    In this question I need to find the inner and outer charge density of the shell I did part A just fine, I used the formula for an electric field due to a line charge, but parts B and C is what's really confusing me. I'm not really sure how to go about it, I placed a spherical gaussian surface...
  2. milkism

    Method of Images, combination of an infinite plane and a hemisphere

    Problem: I have done part a) in spherical polar coordinates. For part b) I thought it would be just: $$\sigma = -\epsilon_0 \frac{\partial V}{\partial r}$$ But I got confused by "You may want to use different coordinate systems .." So I assume partial derivative w.r.t to r is the spherical...
  3. rokiboxofficial Ref

    Surface density of the charges induced on the bases of the cylinder

    The correct answer to this problem is: ##\sigma = \varepsilon_0E\frac{\varepsilon-1}{\varepsilon}## Here is my attempt to solve it, please tell me what is my mistake? ##E_{in} = E_{out} - E_{ind}## ##E_{ind} = E_{out} - E_{in}## ##E_{in} = \frac{E_{out}}{\varepsilon}## ##E_{ind} = E_{out} -...
  4. G

    Surface charge density of a plane

    $$\phi_E=\dfrac{Q_{\textrm{enclosed}}}{\varepsilon_0}\Rightarrow Q_{\textrm{enclosed}}=9,6\cdot 10^{-7}\, \textrm{C}$$ $$Q_{\textrm{enclosed}}=\sigma S=\sigma \pi R^2\Rightarrow \sigma =\dfrac{Q_{\textrm{enclosed}}}{\pi (0,1^2)}=3,04\cdot 10^{-5}\, \textrm{C}/\textrm{m}^2$$ I have a lot of...
  5. patric44

    Confirming the dimension of induced charge density of a dielectric

    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 $$ \rho = \frac{-1}{4\pi k} \vec{E}\cdot\;grad(k) $$ $$ \sigma= \frac{-(k-1)}{4\pi} \vec{E_{1}}\cdot\;\vec{n} $$ I am somehow confused about the...
  6. cwill53

    Capacitor and Surface Charge Density Question

    When I plug in the numbers I get ##2.9513\cdot 10^{-5}C/m^2##, not ##17.6\cdot 10^{-6} C/m^2##. Can someone point out where I'm going wrong?
  7. Samanko

    Surface charge density of a conducting spherical shell

    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?
  8. A

    Variable induced surface charge density on a sphere

    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...
  9. A

    How does the dielectric affect the charge distribution on a charged sphere?

    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??
  10. Dor

    Steady state boundary conditions between metal/dielectric?

    There are few thing I'm not sure of and be happy for clarifications. In general: at steady state, what are the electric-field,potential, and current boundary conditions between a conductor and a dielectric medium? more specific: a) When dealing with a perfect conductor there exist a surface...
  11. T

    Surface Charge density of glass cyclinder

    Homework Statement Consider an infinitely long one dimensional conducting wire with a homeogenous charge density ##\lambda##, running along the central axis of an infinitely long cyclindrical glass casing of radius b (glass is a dielectric material). Calculate: a) The displacement vector...
  12. L

    How to calculate the dipole moment of the spherical shell?

    Homework Statement A spherical shell of radius R has a surface charge distribution σ = k sinφ. Calculate the dipole moment of the spherical shell. Homework Equations P[/B]' = ∫r' σ(r') da' The Attempt at a Solution So I believe my dipole will be directed along the y axis, as the function...
  13. B

    Electric field and surface charge density problem

    Homework Statement Homework EquationsThe Attempt at a Solution I don't understand how it looks like, how to approach this problem. I think it's maybe image charge or Laplace equation
  14. Mictlantecuhtli

    Expressing surface charge density as volume charge density

    I'm working on this: When I consider a disc with radius ##a## and total charge ##Q## uniformly distributed (placed in the XY plane and centered at the origin) and determine the volume charge density in cylindrical coordinates, I have assumed is of the form ##\rho=A \delta (z) U(R-r)##, (##U## is...
  15. Cocoleia

    Electric field, surface charge density

    Homework Statement Homework EquationsThe Attempt at a Solution I have the full solution, the first part being: I don't understand how they came up with the expression for Vab. I know usually ΔV=-∫E dl, but I'm not sure how they found their expression. Can someone explain? Thanks.
  16. S

    Induced surface charge density

    Homework Statement Calculate the surface charge density induced by a point source above an infinite conducting plane, with 0 potential. Homework Equations ##E=-\nabla V## ##V=\frac{q}{4\pi \epsilon_0 r}## The Attempt at a Solution I used the method of image charges and I calculated the...
  17. S

    Induced surface charge density

    Hello! I am a bit confused about calculating the induced surface charge density on an infinite conducting plane, with 0 potential, in the presence of a charge, q, a distance d above it. Assuming that the plane is in xy plane and the charge in positive z region, in the book they use the method of...
  18. Mason Smith

    What is the surface charge density on the face of the penny?

    Homework Statement The electric field strength just above one face of a copper penny is 2230 N/C. What is the surface charge density on this face of the penny? Homework Equations Electric field of an infinite plane of charge = η/(2*ε0) The Attempt at a Solution I used the above equation, and...
  19. J

    What is the Surface Charge Density Based on Velocity Over Time Graph?

    Homework Statement Homework Equations E=σ/(2Eo) σ=2Eoma/e a=Δv/Δt The Attempt at a Solution So doing my best to read the velocity over time graph I came up with Δv/Δt=(-2E5m/s)/(10E-12s)=-2E17 m/s/s σ=2Eoma/e =2(8.99E-9C)(9.109E-31kg)(-2E17m/s/s)/(1.6022E-19) =-0.020444 C/m^2 but by the...
  20. Destroxia

    Surface Charge Density, Polarization

    Homework Statement The electric dipole moment for the water molecule equals $$ p = 6.13 × 10−30 C · m $$ Suppose that in the glass of water all molecular dipoles could be made to point down. Calculate the resulting surface charge density at the upper water surfaceHomework Equations [/B] ## P...
  21. F

    Volumentric or surface charge density

    Homework Statement It is known that the potencial is given as V = 80 ρ0.6 volts. Assuming free space conditions, find a) E, b) the volume charge density at ρ=0.5 m and c) the total charge lying withing the closed surface ρ=0.6, 0<z<1 Homework Equations E[/B]=-∇VThe Attempt at a Solution (this...
  22. T

    A Charged Conductive Plate with a Charged Plane

    Homework Statement In the "Before" scenario, there is conductive plate in electrostatic equilibrium with uniform surface charge density +η (the plate has some thickness but the width and length are significantly larger). In the "After" scenario, an infinite plane of negative charge with fixed...
  23. Titan97

    Surface charge density of conducting disc

    Homework Statement This is problem 3.4 from Prucell and Morin if you have the book. Homework Equations None The Attempt at a Solution Electric field inside a conducting sphere is zero. Let P be a point on one of its equatorial plane. The field along the plane is zero. So I know the charge...
  24. R

    What is the surface current density at a general point on a conducting sphere?

    Homework Statement Electric current of I amperes flows along the z-axis from (0, 0,-∞) to (0, 0, -a) and from there it spreads over a conducting sphere r = a in the -aθ direction, comes to the point (0; 0; a) and goes to (0, 0, ∞) again along the z-axis. What is the surface current density at...
  25. Jimbob999

    Surface Charge Density (Electric Fields)

    A disk with a uniform positive surface charge density lies in the x-y plane, centered on the origin. The disk contains 2.5 x 10-6 C/m2 of charge, and is 7.5 cm in radius. What is the electric field at z = 15 cm? I have used the formula...
  26. P

    Charge density higher at sharp points

    Hi... I want to know why charge density is higher at sharp points in a conductor? I have gone through the analogy of two spheres connected by a wire... But is there any other explanation which is not specific to spheres...?
  27. Estefania_8

    Potential of a Charged Conducting Sphere

    Homework Statement A conducting sphere with radius R is charged to voltage V0 (relative to a point an infinite distance from the sphere where the potential is zero). What is the surface charge density σ? Express your answer in terms of the given quantities and ϵ0. Homework Equations Electric...
  28. O

    Dielectric effects confined inside the dielectric material?

    Hi all, I am wondering how is it possible that the polarization effects of a dielectric material remain confined inside the material itself. That is: for a LIH dielectric, the equations state that the electric field inside the material is reduced by \epsilon_r. But outside the material, no...
  29. A

    E-field and charge density outside two coaxial cylinders

    I am working on a problem of electrostatics, and I am having troubles in trying to figure out one part of it. 1. Homework Statement It consists of an inner solid cylinder of radius ##a## with a voltage ##V_A##, and an outer coaxial cylindrical shell of inner radius ##b## and outer radius ##c##...
  30. Roodles01

    Free surface charge density (not bound density)

    Homework Statement a parallel plate capacitor has 2 plates separated by a dielectric of rel. permittivity 5.0 are separated by 0.20mm and have area of 5.0 cm2. Potential difference between the plates is 500V. I need to be able to calculate the free surface charge density...
  31. amjad-sh

    Non uniform surface charge density

    hello. The problem is : A metallic horizontal disk of radius R has been charged on its top surface with a total charge Q such that its surface charge density is "non uniform" and governed by the expression σ=q|cosθ|*r where r and θ are the polar coordinates measured from the center O of the...
  32. V

    Volume charge density w/o surface charge density

    Im confused by a concept i have run across in Griffiths electrodynamics. E_{out} - E_{in} = \frac{\sigma_{free}}{\epsilon_0} However, in the case of a uniform, circular charge density, \vec{E_{in}} = \frac{\rho r}{3\epsilon_0}\hat{r} \vec{E_{out}} = \frac{\rho R^3}{3\epsilon_0...
  33. Ascendant78

    Why does surface charge density change for an inner surface?

    Rather than try to explain what I am talking about, I am going to link to an image of it below. My question is in regards to the negative value I marked in the answer (the one for the surface charge density for the inside of the outer shell). While I get the concept that it would be negative...
  34. R

    Surface charge density on a cylindrical cavity

    Hi folks, I am having trouble generalizing a well-known problem. Let's say we have a cylindrical cavity inside a conductor, and in this cavity runs a line charge λ. I would now like to know the surface charge density on the inside wall of the cavity, but with the line charge not in the center of...
  35. M

    Finding Surface Charge Densities in a Spherical Conductor

    Homework Statement A point charge +Q is placed at the center of a spherical insulator of radius a. The insulator completely fills three cavity of a spherical conducting shell of radius b. Find the inner and outer surface charge density of the conductor and the bound surface charge density of...

    When does the surface charge density equal the electric flux density?

    Homework Statement The electric field intensity in polystyrene (relative permittivity 2.55) filling the space between the plates of a parallel plate capacitor is 10kV/m. The distance between the plates is 1.5mm. Calculate the surface charge density of free charge on the plates, and the surface...
  37. Z

    Surface Charge Density - notation used

    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.
  38. N

    Surface charge density due to point charge

    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...
  39. T

    Consider a spherical shell with radius R and surface charge density σ = σ0 cosθ

    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...
  40. P

    How to determine the surface charge density for a parallel plate capacitor

    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...
  41. P

    Surface Charge Density and electric field

    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...
  42. M

    Find the surface charge density of a perfect conductor

    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...
  43. S

    What is the surface charge density?

    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...
  44. I

    Surface Charge Density of a Neutral Conducting Cube

    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...
  45. J

    Surface charge density on conducting cone with point charge inside

    Homework Statement Calculate the surface charge density on a thin insulated and uncharged cone, which has a point charge inside of it on the cone axis. Furthermore, calculate the force between the point charge and the cone. Homework Equations The relevant equation is the Poisson equation...
  46. G

    Finding an electric field on a disk using surface charge density

    Homework Statement A disk of radius 2.30 cm has a surface charge density of 5.22 μC/m2 on its upper face. What is the magnitude of the electric field produced by the disk at a point on its central axis at distance z = 11.2 cm from the disk? Homework Equations Formula for an electric...
  47. P

    Find the surface charge density that will stop a proton

    Homework Statement 2. The attempt at a solution I have the solution, but it's pretty confusing so I want to see if anyone else can walk me through step by step. The first step is pretty clear...You have to find the acceleration that the proton will experience as it moves towards the...
  48. P

    Ohm's Law: Coaxial Cylinder Resistance & Density

    Homework Statement Consider two coaxial conducting cylinders with radii a and 3a and length L. The region a<r<2a is filled with a material of conductivity σ1, and the region 2a<r<3a has conductivity σ2. (Assume ε1=ε2=εo.) The inner cylinder is held at potential V0 and the outer cylinder at...