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Physics we place a small planar element of area S normal to E at a point, the number of field lines crossing it is proportional* to E S. Now suppose we tilt the area element by angle . Clearly, the number of field lines crossing the area element will be smaller. The projection of the area element normal to E is S cos . Thus, the number of field lines crossing S is proportional to E S cos . When = 90°, field lines will be parallel to S and will not cross it at all (Fig. 1.15). The orientation of area element and not merely its magnitude is important in many contexts. For example, in a stream, the amount of water flowing through a ring will naturally depend on how you hold the ring. If you hold it normal to the flow, maximum water will flow FIGURE 1.15 Dependence of flux on the inclination between E and n. through it than if you hold it with some other orientation. This shows that an area element should be treated as a vector. It has a magnitude and also a direction. How to specify the direction of a planar area? Clearly, the normal to the plane specifies the orientation of the plane. Thus the direction of a planar area vector is along its normal. How to associate a vector to the area of a curved surface? We imagine dividing the surface into a large number of very small area elements. Each small area element may be treated as planar and a vector associated with it, as explained before. Notice one ambiguity here. The direction of an area element is along its normal. But a normal can point in two directions. Which direction do we choose as the direction of the vector associated with the area element? This problem is resolved by some convention appropriate to the given context. For the case of a closed surface, this convention is very simple. The vector associated with every area element of a closed surface is taken to be in the direction of the outward normal. This is the convention used in Fig. 1.16. Thus, the area element vector S at a point on a closed surface equals S n where S is the magnitude of the area element and nˆ is a unit vector in the direction of outward normal at that point. We now come to the definition of electric flux. Electric fluxthrough an area element S is defined by . = E S = E S cos (1.11) which, as seen before, is proportional to the number of field lines cutting the area element. The angle here is the angle between E and S. For a closed surface, with the convention stated already, is the angle between FIGURE 1.16 E and the outward normal to the area element. Notice we could look at Convention for the expression E S cos in two ways: E ( S cos ) i.e., E times the defining normal n and S. * It will not be proper to say that the number of field lines is equal to E S. The number of field lines is after all, a matter of how many field lines we choose to draw. What is physically significant is the relative number of field lines crossing 22 a given area at different points. Reprint 2026-27