Discuss nature of electric flux in a non uniform electric field and arbitrarily shaped area.

Explanation

If the field is not uniform and the area A is not flat (figure), then the entire area is divided into n small area segments `DeltavecA_1,DeltavecA_2,DeltavecA_2......DeltavecA_n` such that each area element is almostm flat and the electric field through each area element is considered to be uniform.
The electric flux for the entire area A is approximately written as `phi_E = vecE_1*DeltavecA_1+vecE_2*DeltavecA_2+vecE_3*DeltavecA_3......vecE_n*DeltavecA_n`
`= sum_(i=1)^nvecE*DeltavecA_1` ...(1)

By taking the limit `DeltaA_i rarr 0` (for all i) the summation in equation (1) becomes integration. The total electric flux for the entire area is give by
`phi_E = intvecE.dvecA` ....(2)
From Equation (2), it is clear that the electric flux for a given surface depends on both the electric field pattern on the surface area and orientation of the surface with respect to the electric field.