Continuous Charge Distribution
As charge can exist only as integral multiple of basic charge (e) therefore; charge distribution is always discrete, on account of atomicity of charge. However, it is impractical to work in terms of discrete charges always. For example, on the surface of a charged conductor we cannot specify the charge distribution in terms of the locations of the microscopic charged constituents. However, we can consider a small area element Δ S on the surface of the conductor; this area element is very small on the macroscopic scale, but big enough to include a very large number of electrons. If Δ Q is the amount of charge on this element, we define surface charge density (σ) at the area element by
σ =Δ Q / ΔS
We can repeat the process at different points on the surface of the conductor and thus arrive at a continuous functionσ, called the surface charge density. At the microscopic level, charge distribution is discontinuous, as there are discrete charges separated by intervening space where there is no charge. Therefore, σ represents macroscopic surface charge density which is a smoothed out average of the microscopic charge density over an area element Δ S – which is small macroscopically but large microscopically.
On the same basic when charge is distributed along a line straight or curved, we define linear charge density
Λ = Δ Q / Δ I
Where ?Iis a small line element of wire on the macroscopic scale that includes a large number of microscopic charged constituents and ?Q is the charge contained in that line element. The units of λ are c/m the volume charge density is defined in a similar manner as
P = Δ Q / Δ V
Where Δ Q is the charge included in the macroscopically small volume element Δ V that includes a large number of microscopic charged constituents. The units of p are c/m
Note that the notion of continuous charge distribution is similar to the continuous mass distribution in mechanics. For example when we talk of density of a liquid we are referring to its macroscopic density treating it as a continuous fluid and ignoring its discrete molecular constitution.
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