Continuous Charge Distributions

There may be three types of charge distributions, namely, linear charge, surface charge and volume charge distribution, namely, linear charge, surface charge and volume charge distributions.

(i)    Line Charge. Let λ be the chargers per unit length. If charge is distributed over a line, each differential charge dQ along the line produces a differential electric field.
dE = dQ/4πε₀r² r at P.

Then the total electric field
    E(P) = 1/4πε₀ ∫r/r² λdl                                    … (1)



(ii)    Surface Charge. Charge may also be distributed over a surface or sheet. Let σ be the charge per unit area. Then dQ = σ dσ da where da is the small element of surface area.

Each differential charge dQ on the surface produces a differential electric field
        dE = dQ/4πε₀r² r at point P.

Then the total electric field at P is
        E (P) = 1/4πε₀ ∫r/r² σda.                                … (2)


(iii)    Volume Charge. Charge spread throughout a volume is described by ρ (the charge per unit volume). Then dQ = ρ dτ is the small element of volume. Each differential charge dQ produces a differential charge dQ produces a differential electric filed.
dE = dQ /4πε₀r² r at point P.)

The total electric field at P is obtained by integration over the volume:
        E (P) = 1/4πε₀ ∫r/r²ρdτ

In Eqs. (1), (2) and (3), r is the distance from the element of charge (therefore, from dl, da, or dτ) to the point P.