Explain the nature ofelectric flux for uniform electric field.

Explanation

Let us consider a uniform electric field in a region of space. Let us select an area A normal to the electric field lines. (Fig. (a). The electric flux for this case is

`phi_E = EA` …(1)
Here `vecA = Ahatn`

If the same area A is kept parallel to the uniform electric field, then no electric field lines pierce through the area A. (Fig. (b)). The electric flux for this cas is zero.
`pih_E = 0` ...(2)
If the area is inclined at an angle `theta` with the field, then the component of the electric field perpendicular to the area alone contributes to the electric flux. The electric field component parallel to the surface area will not contribute to the electric flux
`phi_E = (E costheta)A` ...(3)

Further, theta is also the angle between the electric field and the direction normal to the area. Hence in general, for uniform electric field, the electric flux is defined as
`phi_E = vecE.vecA = EAcostheta` ...(4)
Here, note that `vecA` is the area vector `vecA = Ahatn`. Its magnitude is simply the area A and the direction is along the unit vector `hatn` perpendicular to the area. Using ths definition for flux, `phi_E = vecE.vecA`, equations (2) and (3) can be obtained as special cases.
In figure (a), `theta=0^@` so `phi_E = vecE.vecA = EA`
In figure (b), `theta=90^@` so `phi_E = vecE.vecA = 0`