Consider the following charged object of irregular object of irregular shape as shown in Figure. The entire charged object is divided into a large number on charge elements `Deltaq_(1),Deltaq_(2), Deltaq_(3),………Deltaq_(n)`, and each charge element `Deltaq` is taken as a point charge.
(ii) The electric field at a point P due to a charge object is approximately given by the sum of the fields at P due to all such charge elements.
`vec(E)=(1)/(4piepsi_(0))((Deltaq_(1))/(r_(1P)^(2))hatr_(1P)+(Deltaq_(2))/(r_(2P)^(2))hatr_(2P)+..........+(Deltaq_(n))/(r_(nP)^(2))hatr_(nP))`
`=(1)/(4piepsi_(0))sum_(t=1)^(n)(Deltaq_(i))/(r_(iP)^(2))hatr_(iP)`
Here `Deltaq_(i)` is the `i^(th)` charge element, `r_(iP)` is the distance of the point P from the `i^(th)` charge element and `hatr_(iP)` is the unit vector from the `i^(th)` charge element to the piont P. However the equation (1) is only an approximation. To incorporate the continuous distribution of charge, we take the limit `Deltaqto0(=dq)`. In this limits, the summation in the equation (1) becomes an integration and takes the following form
`vec(E)=(1)/(4piepsi_(0))int(dq)/(r^(2))hatr" "....(2)`
(iv) Here r is the distance of the point P from the infinitesimal charge dq and `hatr` is the unit vector from dq to point P. Even though the electric field for a continuous charge distribution can be difficult to evaluate, the force experienced by some test charge q in this electric field is still given by `vec(F)=qvec(E)`
(a) If the charge Q is uniformly distributed along the line of length L, then linear charge density (charge per unit length) `lamda` is `lamda=(Q)/(L)` unit is coulomb per meter `(Cm^(-1))`. The charge present in the infinitestimal length dl is `dq=lamdadl`. This is shown in Figure 1 (a)
The electric field due to the line of total charge Q is given by
`vec(E)=(1)/(4piepsi_(0))int(lamdadl)/(r^(2))hatr=(lamda)/(4piepsi_(0))int(dl)/(r^(2))hatr`
If the charge Q is uniformly distributed on a surface of area A, then surface charge denisty (charge per unit area) `sigma` is `sigma=(Q)/(A)`. Is unit is coulomb per square meter `(Cm^(-2))`. The charge present in the infinitesimal area dA is `dq=sigmadA`. This is shown in the figure 1(b). The electric field due to a total charge Q is given by
`vec(E)=(1)/(4piepsi_(0))int(sigmada)/(r^(2))hatr=(1)/(4piepsi_(0))sigmaint(da)/(r^(2))hatr`
This is shown in Figure 1 (b).
If the charge Q is uniformly distributed in a volume V, then volume charge density (charge per unit volume) `pho` is given by `rho=(Q)/(V)`. Its unit is coulomb per cubic meter `(Cm^(-3))`.
The charge present in the infinitesimal volume element dV is `dq=rhodV`. This is shown in Figure 1(c). The electric field due to a volume of total charge Q is given by
`vec(E)=(1)/(4piepsi_(0))int(rhodV)/(r^(2))hatr=(1)/(4piepsi_(0))rhoint(dV)/(r^(2))hatr`.
