What is linear charge density: Definition and 19 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 even smaller scales, of atoms and molecules, 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. C

    Calculating Linear Charge Density of a Cylinder

    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...
  2. S

    Find the linear charge density lambda and the radius of a charged semicircle

    Electric field for the semi-circle $$E = - \frac {πKλ} {2R} $$ 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...
  3. M

    Calculate linear charge density of rod & mag. of E-field at point

    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...
  4. Eclair_de_XII

    More linear charge density troubles....

    Homework Statement An infinitely long line of charge has linear charge density ##λ=4.00_{10^{−12}} \frac{C}{m}##. A proton (mass ##m_p=1.67_{10^{-27}}kg##, charge ##e=1.602_{10^{-19}}C##) is ##r_a=0.18m## from the line and moving directly toward the line at ##v=1000\frac{m}{s}##. Homework...
  5. Eclair_de_XII

    What is wrong with my linear charge density calculations?

    Homework Statement "A straight, nonconducting plastic wire ##x=9.50_{10^{-2}}m## long carries a charge density of ##λ=1.3_{10^{-7}} C/m## distributed uniformly along its length. It is lying on a horizontal tabletop. If the wire is now bent into a circle lying flat on the table, find the...
  6. K

    Linear Charge Density

    Homework Statement This is a wire whose shape is given by y = acos(x/L). This wire has a linear charge density of +λ, and is it desired to determine the electric field at the point (0,y) where y > a. a) If a=0, determine the amount of charge the wire has. b)If a > 0, is the total charge on...
  7. acdurbin953

    Relating Volume Charge Density and Linear Charge Density

    Homework Statement An infinite line of charge with linear density λ1 = 6.7 μC/m is positioned along the axis of a thick insulating shell of inner radius a = 2.4 cm and outer radius b = 4.7 cm. The insulating shell is uniformly charged with a volume density of ρ = -722 μC/m3. What is λ2, the...
  8. B

    Shell's Linear Charge density (electric field mistake?)

    Homework Statement Figure 23-37a shows a narrow charged solid cylinder that is coaxial with a larger charged cylindrical shell. Both are noncon-ducting and thin and have uniform surface charge densities on their outer surfaces. Figure 23-37b gives the radial component E of the electric field...
  9. B

    What is the wire's linear charge density from a proton?

    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...
  10. A

    Help with linear charge density and flux

    Homework Statement The y-axis carries a uniform linear charge density of -2 nC/m, and there is a 8 nC point charge at the point (3 cm, 0 cm, 0 cm) as well as a -4 nC point charge at the point (-8 cm, 0 cm, 0 cm). What is the electric flux through a closed spherical surface of radius 4 cm...
  11. D

    Calculating Maximum Linear Charge Density in Geiger Tube

    Homework Statement For your senior project, you are designing a Geiger tube for detecting radiation in the nuclear physics laboratory. This instrument will consist of a long metal cylindrical tube that has a long straight metal wire running down its central axis. The diameter of the wire...
  12. S

    What is the wire's linear charge density?

    Homework Statement A proton orbits a long charged wire, making 1.60 * 10^6 revolutions per second. The radius of the orbit is 1.60 cm. What is the wire's linear charge density? Homework Equations F=qE F=ma F=(mw^2)/r F=Eklambda/2r=mw^2/r (the radiuses cancel out) The Attempt at...
  13. B

    Finding electric field with a changing linear charge density

    Homework Statement a thin rod of length L is set along an X axis. we want to find the electric field at a point P at the origin, a distance "d" from the rod. The linear change density changes with X and it's given by λ=λ0 ((x-d)^3)/d^3 find the electric field at the point P...
  14. E

    Electric field due to a Linear charge density

    Homework Statement Two long, thin parallel rods, a distance 2b apart, are joined by a semicircular piece of radius b, as shown. Charge of uniform linear density \lambda is deposited along the whole filament. Show that the field E of this charge distribution vanishes at the point C. DO...
  15. L

    Electromagnetism linear charge density question

    Homework Statement Two rods, of lengths l1 and l2, have charges q1 and q2 a)Find the charges per unit length for each rod individually. b)Find the charge per unit length, averaged over both rods. c)Check your result for l1 approaching 0 d)Check your result for l1=l2 Homework...
  16. E

    Linear Charge Density (Question on the problem)

    Homework Statement A charge of -310e is uniformly distributed along a circular arc of radius 4.15 cm, which subtends an angle of 43°. What is the linear charge density along the arc in C/m? Homework Equations Density = Charge/length The Attempt at a Solution Charge =...
  17. C

    Gauss' Law using linear charge density

    Homework Statement A charge of uniform linear density 2.80 nC/m is distributed along a long, thin, nonconducting rod. The rod is coaxial with a long conducting cylindrical shell (inner radius = 5.20 cm, outer radius = 10.8 cm). The net charge on the shell is zero. (a) What is the magnitude...
  18. D

    Electric Force, linear charge density

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