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www.GOALias.blogspot.com26Physicswe place a small planar element of area ΔSnormal to E at a point, the number of field linescrossing it is proportional* to E ΔS. Nowsuppose we tilt the area element by angle θ.Clearly, the number of field lines crossing thearea element will be smaller. The projection ofthe area element normal to E is ΔS cosθ. Thus,the number of field lines crossing ΔS isproportional to E ΔS cosθ. When θ = 90°, fieldlines will be parallel to ΔS and will not cross itat all (Fig. 1.18).The orientation of area element and notmerely its magnitude is important in manycontexts. For example, in a stream, the amountof water flowing through a ring will naturallydepend on how you hold the ring. If you holdit normal to the flow, maximum water will flowFIGURE 1.18 Dependence of flux on theinclination θ between E andthrough it than if you hold it with some otherˆn .orientation. This shows that an area elementshould be treated as a vector. It has amagnitude and also a direction. How to specify the direction of a planararea? Clearly, the normal to the plane specifies the orientation of theplane. 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 imaginedividing the surface into a large number of very small area elements.Each small area element may be treated as planar and a vector associatedwith it, as explained before.Notice one ambiguity here. The direction of an area element is alongits normal. But a normal can point in two directions. Which direction dowe choose as the direction of the vector associated with the area element?This problem is resolved by some convention appropriate to the givencontext. For the case of a closed surface, this convention is very simple.The vector associated with every area element of a closed surface is takento be in the direction of the outward normal. This is the convention usedin Fig. 1.19. Thus, the area element vector ΔS at a point on a closedsurface equals ΔS ˆn where ΔS is the magnitude of the area element andˆn is a unit vector in the direction of outward normal at that point.We now come to the definition of electric flux. Electric flux Δφ throughan area element ΔS is defined byΔφ = E.ΔS = E ΔS cosθ (1.11)FIGURE 1.19Convention fordefining normalwhich, as seen before, is proportional to the number of field lines cuttingthe area element. The angle θ here is the angle between E and ΔS. For aclosed surface, with the convention stated already, θ is the angle betweenE and the outward normal to the area element. Notice we could look atthe expression E ΔS cosθ in two ways: E (ΔS cosθ ) i.e., E times theˆn and ΔS. * It will not be proper to say that the number of field lines is equal to EΔS. Thenumber of field lines is after all, a matter of how many field lines we choose todraw. What is physically significant is the relative number of field lines crossinga given area at different points.