Surface charge density Definition and 86 Threads

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.

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  1. F

    How Do You Calculate Surface Charge Density Without Knowing Plate Dimensions?

    Homework Statement [PLAIN]http://img232.imageshack.us/img232/8974/61038710.png The Attempt at a Solution Well \sigma = charge/area But how am I suppose to find the area if I am not even given the dimensions of the plates?
  2. M

    Minimum Surface Charge Density for Spark Creation

    Homework Statement A spark occurs at the tip of a metal needle if the electric field strength exceeds 3.0x10^6 N/C. What is the minimum surface charge density for producing a spark? Homework Equations int(EdA) = (Qin) / e0 The Attempt at a Solution I imagine the surface as a cylinder with...
  3. M

    Finding the surface charge density of an infinite sheet next to a dipole

    Homework Statement An infinite, vertical, nonconducting plane sheet is uniformly charged with electricity. Next to the sheet is a dipole that can freely oscillate about its midpoint O, which is at a distance Ж from the sheet. Each end of the dipole bears a charge q and a mass m. The length of...
  4. E

    Surface Charge Density for Infinite Sheet of Charge

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  5. B

    Potential inside sphere with surface charge density

    Homework Statement A specified charge density sigma(theta)=kcos(theta) is glued on the surface of a spherical shell of radius R. find the resulting potential inside and outside of the sphere. Homework Equations The Attempt at a Solution This is a worked example in Griffiths. The...
  6. P

    Surface Charge Density of shower

    Homework Statement When a shower is turned on in a closed bathroom, the splashing of the water on the bare tub can fill the room's air with negatively charged ions and produce an electric field in the air as great as 1000 N/C. Consider a bathroom with dimensions 2.3 m x 3.6 m x 1.6 m. Along...
  7. D

    Cylindrical Surface Charge Density

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  8. P

    How Is Surface Charge Density Calculated from Electric Field Magnitude?

    Homework Statement A large plane having uniform charge density has an electric field just outside the plane that points directly toward the plane with magnitude 1.50×10^4 N/C . What is the surface charge density? Express your answer with the appropriate unit Homework Equations I am...
  9. S

    Point charge above xy plane, find surface charge density

    Homework Statement A point charge, Q, is located @ (0,0,d) above infinite conducting plane that lies in xy plane and is maintained at ground potential. Find: a.) surface charge density as a function of x and y on conducting plane, and b.) total charge induced on conducting plane...
  10. L

    Surface charge density problem

    1. A spark occurs at the tip of a metal needle if the electric field strength exceeds 4.00×106 N/C, the field strength at which air breaks down. What is the minimum surface charge density for producing a spark? 2. surface charge density = Q/A 3. The answer is 3.54*10-5 C/m2. How was this...
  11. C

    What is the Surface Charge Density for a Spherical Cavity?

    Homework Statement A sphere with excess charge of 1.5 x10 -7 Coulombs, An outer radius of 2.5cm and inner radius of 1.5 cm ( a cavity ). The question is to find the surface charge density on the inside surface. I'm totally at a loss on this one and couldn't find how to solve it...
  12. R

    Surface Charge Density on an infinite sheet

    A small, nonconducting ball of mass 1.4E-6 kg and charge 1.9E-8 C hangs from an insulating thread that makes an angle of 32 degrees with a vertical, uniformly charged nonconducting sheet. Considering the gravitational force on the ball and assuming the sheet extends far vertically and into and...
  13. F

    Surface charge density and more, for a parallel plate cap.

    Homework Statement An air-filled capacitor consists of two parallel plates, each with an area of 7.60 cm2, separated by a distance of 1.60 mm. 1. If a 25.0 V potential difference is applied to these plates, calculate the electric field between the plates. 2. What is the surface charge density...
  14. S

    Gauss' Law & Surface Charge Density

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  15. J

    Calculate Dipole Moment of Spherical Shell w/ Surface Charge Density

    Calculate the dipole moment of a spherical shell of radius 'a' whose surface charge density is σ=σ0(1+cos(θ)). The origin is at the center of the sphere. What I know: 1) p=∫r'*ρ*dV' I'm having trouble understanding how to transform this equation so that I can calculate the dipole moment...
  16. K

    Force on a point charge due to constant sphere surface charge density.

    Homework Statement The surface of a sphere of radius a is charged with a constant surface density \sigma. What is the total charge Q' on the sphere? Find the force produced by this charge distribution on a point charge q located on the z axis for z > a and for z < a.Homework Equations The...
  17. J

    How Can Surface Charge Density Equal bcos(θ)?

    Homework Statement A surface is defined by a hemisphere of radius b, centered on the x-y plane. The surface charged density is given by \rho_s(z) = z (\frac{Coul}{m^3}). Homework Equations \rho_s(z) = z = Rcos(\theta) = bcos(\theta) (\frac{Coul}{m^3}). 3. Question My question is...
  18. B

    Compute the surface charge density on the belt

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  19. J

    Finding the surface charge density

    Homework Statement A 1.0-mm-diameter wire has 1000 excess electrons per centimeter of length. What is the surface charge density? Homework Equations \eta = Q/A The Attempt at a Solution \eta = (100000 * 1.6*10^-19)/(\pi * (5*10^-4)2) But that gets me 2.037 * 10^-8 C/m^2 The...
  20. L

    Finding the surface charge density of plates in order to stop a proton

    Homework Statement You have a summer intern position at a laboratory that uses high speed proton beam. The protons exit the machine at a speed of 2.0*10^6 m/s [...] You decide to slow the protons to an acceptable speed (2.0*10^5 m/s), then let them hit a target. you take two metal plates...
  21. T

    Convergence of the surface charge density Fourier series expansion

    Homework Statement Test the convergence of the series for the surface charge density: \sum^{\infty}_{s=0}(-1)^s(4s+3)\frac{(2 s -1)!}{(2s)!} Homework Equations (2s-1)! = \frac{(2s)!}{2^s s!}; (2s)! = 2^s s! Stirling's asymptotic formula for the factorials: s! = \sqrt{2 \pi s}s^s...
  22. M

    Non-convergent series for surface charge density

    1. This is a 2D Laplace eqn problem. A semi-infinite strip of width a has a conductor held at potential V(0,a) = V_0 at one end and grounded conductors at y=0 and y=a. Find the induced surface charge \sigma (y) on the conductor at x=0. 2. Homework Equations . The potential is...
  23. A

    Calculating Surface Charge Density

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  24. K

    How to calculate the flux given the surface charge density?

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  25. G

    Surface Charge Density and electron

    Homework Statement an electron with mass m = 9.11 x 10^-31 is released at rest near a very large positively charged non-conducting charged sheet lying horizontally. What should the surface charge density on this sheet be to keep the electron balanced at rest above the ground? Is it above or...
  26. A

    Why is the electric field infinite at the location of a charge?

    At a point where there's surface charge density sigma, by what amount is the electric field for a spherical shell of charge discontinuous? I thought about using E = 2*pi*k*sigma (although isn't that just true for a charged disc?)
  27. F

    Damn units and surface charge density.

    Ok, the question asks: An electron close to a large, flat sheet of charge is repelled from the sheet with a 2.1×10^−12 {N} force. Q. find the surface charge density on the sheet in C/m^2 What i did was use the formula F = qe with 2.1e-12 for the force and 6.1e-19 for the charge of an...
  28. N

    Finding magnitude of surface charge density.

    If given variables A, d, V, and constant 8.854e-12. How can I find the magnitude of surface charge density. I originally used Sigma = Q/A but I am not getting the right answer. Any help is appreciated! :)
  29. D

    What is the surface charge density sigma?

    [SOLVED] Electric Potential Homework Statement 20. [1pt] A 30.6- cm-diameter conducting sphere is charged to 538 V relative to V = 0 at r =infinity. What is the surface charge density sigma? Correct, computer gets: 3.11E-08 C/m^2 = Sigma, r=.153 21. [1pt] At what distance will the potential...
  30. G

    Surface charge density of Styrofoam

    A 10.0g piece of styrofoam carries a net charge of -.700 x 10^-6 C and floats above the center of a large horizontal sheet of plastic that has a uniform charge density on its surface. What is the charge per unit area on the plastic sheet? I've been trying to think a way to model this with a...
  31. S

    Surface Charge Density and high-speed proton beam

    You have a summer intern position at a laboratory that uses a high-speed proton beam. The protons exit the machine at a speed of 2.10 times 10^6 m/s, and you've been asked to design a device to stop the protons safely. You know that protons will embed themselves in a metal target, but protons...
  32. S

    Calculating Required Surface Charge Density for a Cylinder

    Homework Statement A long, straight wire has a linear charge density of magnitude 3.6nC/m. The wire is to be enclosed by a thin, no-conducting cylinder of ouside radius 1.5cm, coaxil witht he wire. The cylinder is to have positive charge on its outside surface with a surface charge density...
  33. J

    What is the surface charge density of the plane?

    I have two problems that I need help working on. The first one is An electron is released from rest 2.0 cm from an infinite charged plane. It accelerates toward the plane and collides with a speed of 1.0 x 10^7 m/s. What is the surface charge density of the plane? The second problem is: A...
  34. T

    Solve Surface Charge Density & He Atom Potential Energy Questions

    I still don't understand from the msg given... another explanation any1 Hey I have two questions... the first one I did I got wrong but am really unsure why... I used sigma=Q/A but didnt get the right answer... I got that the answer was 1.3277E7 but I am so far off it aint funny! Please...
  35. S

    Calculating Surface Charge Density in a Uniform Electric Field

    An electron remains stationary in na electric field directed downward in teh Earth's gravitational field. If the electric field is due to charge on the two large conducting plates oppositely charged and separated by 2.3cm what is the surface charge density assumed to be uniform on the plates...
  36. M

    Find the surface charge density

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