field decreases as the square of the distance from a point charge and the area enclosing the charge increases as the square of the distance, the number of field lines crossing the enclosing area remains constant, whatever may be the distance of the area from the charge. We started by saying that the field lines carry information about the direction of electric field at different points in space. Having drawn a certain set of field lines, the relative density (i.e., closeness) of the field lines at different points indicates the relative strength of electric field at those points. The field lines crowd where the field is strong and are spaced apart where it is weak.
Figure . shows a set of field lines. We can imagine two equal and small elements of area placed at points R and S normal to the field lines there. The number of field lines in our picture cutting the area elements is proportional to the magnitude of field at these points.
The picture shows that the field at R is stronger than at S. To understand the dependence of the field lines on the area, or rather the solid angle subtended by an area element, let us try to relate the area with the solid angle, a generalization of angle to three dimensions. Recall how a (plane) angle is defined in two-dimensions. Let a small transverse line element Δ l be placed at a distance r from a point O.
Then the angle subtended by Δ l at O can be approximated as Δθ = Δ l / r . Likewise, in three-dimensions the solid angle * subtended by a small perpendicular plane area Δ S , at a distance r , can be written as ΔΩ = Δ S / r . We know that in a given solid angle the number of radial field lines is the same. In Fig.
. , for two points P and P at distances r and r from the charge, the element of area subtending the solid angle ΔΩ is r