The number of electric field lines crossing a given area kept normal to the electric field lines is called electric flux. It is usually denoted by the Greek letter ΦE and its unit is N m2 C-1. Electric flux is a scalar quantity and it can be positive or negative. For a simpler understanding of electric flux, the following Figure 1.32 is useful.
The electric field of a point charge is drawn in this figure. Consider two small rectangular area elements placed normal to the field at regions A and B. Even though these elements have the same area, the number of electric field lines crossing the element in region A is more than that crossing the element in region B. Therfore the electric flux in region A is more than that in region B. The electric field strength for a point charge decreases as the distance increases, then for a point charge electric flux also decreases as the distance increases. The above discussion gives a qualitative idea of electric flux. However a precise definition of electric flux is needed.
Consider a uniform electric field in a region of space. Let us choose an area A normal to the electric field lines as shown in Figure 1.33 (a). The electric flux for this case is
Suppose the same area A is kept parallel to the uniform electric field, then no electric field lines pierce through the area A , as shown in Figure 1.33(b). The electric flux for this case is zero.
If the area is inclined at an angle θ 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. This is shown in Figure 1.33 (c). For this case, the electric flux
Further, θ 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
Here, note that is the area vector =A. Its magnitude is simply the area A and the direction is along the unit vector n perpendicular to the area as shown in Figure 1.33.Using this definition for flux, ΦE = ⋅ , equations (1.53) and (1.54) can be obtained as special cases.
In Figure 1.33 (a), θ = 0º so ΦE = ⋅ = EA
In Figure 1.33 (b), θ = 90 º so ΦE = ⋅ = 0
Calculate the electric flux through the rectangle of sides 5 cm and 10 cm kept in the region of a uniform electric field 100 NC-1. The angle θ is 60o. Suppose θ becomes zero, what is the electric flux?
The electric flux
Suppose the electric field is not uniform and the area A is not flat (Figure 1.34), then the entire area is divided into n small area segments ∆1 , ∆2 , ∆3 ......... ∆n suchthat each area element is almost flat and the electric field over each area element is considered to be uniform.
The electric flux for the entire area A is approximately written as
By taking the limit ∆i → 0 (for all i) the summation in equation (1.56) becomes integration. The total electric flux for the entire area is given by
From Equation (1.57), 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.