CONTINUOUS CHARGE DISTRIBUTION
We have so far dealt with charge configurations involving discrete charges `q_1, q_2, ..., q_n.` For example, on the surface of a charged conductor, it is impractical to specify the charge distribution in terms of the locations of the microscopic charged constituents. It is more feasible to consider an area element ΔS (Fig) on the surface of the conductor (which is very small on the macroscopic scale but big enough to include a very large number of electrons) and specify the charge ΔQ on that element. We then define a surface charge density σ at the area element by
`sigma = (DeltaQ)/(DeltaS)`
We can do this at different points on the conductor and thus arrive at a continuous function σ, called the surface charge density. The surface charge density σ so defined ignores the quantisation of charge and the discontinuity in charge distribution at the microscopic level. σ represents macroscopic surface charge density, which in a sense, is a smoothed out average of the microscopic charge density over an area element ΔS which, as said before, is large microscopically but small microscopically. The units for σ are `C/m^2.`
Similar considerations apply for a line charge distribution and a volume charge distribution. The linear charge density λ of a wire is defined by
`lambda = (DeltaQ)/(Deltal)`
where Δl is a small line element of wire on the macroscopic scale that,however, includes a large number of microscopic charged constituents,
and ΔQ is the charge contained in that line element. The units for λ are `C/m.` The volume charge density (sometimes simply called charge density) is defined in a similar manner:
`rho = (DeltaQ)/ (DeltaV)`
where ΔQ is the charge included in the microscopically small volume element ΔV that includes a large number of microscopic charged
constituents. The units for ρ are `C/m^3.`
`sigma = (DeltaQ)/(DeltaS)`
We can do this at different points on the conductor and thus arrive at a continuous function σ, called the surface charge density. The surface charge density σ so defined ignores the quantisation of charge and the discontinuity in charge distribution at the microscopic level. σ represents macroscopic surface charge density, which in a sense, is a smoothed out average of the microscopic charge density over an area element ΔS which, as said before, is large microscopically but small microscopically. The units for σ are `C/m^2.`
Similar considerations apply for a line charge distribution and a volume charge distribution. The linear charge density λ of a wire is defined by
`lambda = (DeltaQ)/(Deltal)`
where Δl is a small line element of wire on the macroscopic scale that,however, includes a large number of microscopic charged constituents,
and ΔQ is the charge contained in that line element. The units for λ are `C/m.` The volume charge density (sometimes simply called charge density) is defined in a similar manner:
`rho = (DeltaQ)/ (DeltaV)`
where ΔQ is the charge included in the microscopically small volume element ΔV that includes a large number of microscopic charged
constituents. The units for ρ are `C/m^3.`