1.14 Consider a uniform electric field E = 3 × 103 î N/C. (a) What is the flux of this field through a square of 10 cm on a side whose plane is parallel to the yz plane? (b) What is the flux through the same square if the normal to its plane makes a 60° angle with the x-axis?

1.18 A point charge of 2.0 mC is at the centre of a cubic Gaussian surface 9.0 cm on edge. What is the net electric flux through the surface?

1.9 ELECTRIC FLUX Consider flow of a liquid with velocity v, through a small flat surface dS, in a direction normal to the surface. The rate of flow of liquid is given by the volume crossing the area per unit time v dS and represents the flux of liquid flowing across the plane. If the normal to the surface is not parallel to the direction of flow of liquid, i.e., to v, but makes an angle q with it, the projected area in a plane perpendicular to v is δ dS cos q. Therefore, the flux going out of the surface dS is v. ˆn dS. For the case of the electric field, we define an analogous quantity and call it electric flux. We should, however, note that there is no flow of a physically observable quantity unlike the case of liquid flow. In the picture of electric field lines described above, FIGURE 1.14 Field lines due to we saw that the number of field lines crossing a unit area, some simple charge configurations. placed normal to the field at a point is a measure of the strength of electric field at that point. This means that if 21 Reprint 2025-26 Physics we place a small planar element of area DS normal to E at a point, the number of field lines crossing it is proportional* to E DS. Now suppose we tilt the area element by angle q. Clearly, the number of field lines crossing the area element will be smaller. The projection of the area element normal to E is DS cosq. Thus, the number of field lines crossing DS is proportional to E DS cosq. When q = 90°, field lines will be parallel to DS 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 through it than if you hold it with some other inclination q between E and ˆn . 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 DS at a point on a closed surface equals DS ˆn where DS is the magnitude of the area element and ˆn is a unit vector in the direction of outward normal at that point. We now come to the definition of electric flux. Electric flux Df through an area element DS is defined by Df = E.DS = E DS cosq (1.11) which, as seen before, is proportional to the number of field lines cutting the area element. The angle q here is the angle between E and DS. For a closed surface, with the convention stated already, q is the angle between E and the outward normal to the area element. Notice we could look at FIGURE 1.16 the expression E DS cosq in two ways: E (DS cosq ) i.e., E times the Convention for defining normal ˆn and DS. * It will not be proper to say that the number of field lines is equal to EDS. 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 2025-26 Electric Charges and Fields projection of area normal to E, or E^ DS, i.e., component of E along the normal to the area element times the magnitude of the area element. The unit of electric flux is N C–1 m2. The basic definition of electric flux given by Eq. (1.11) can be used, in principle, to calculate the total flux through any given surface. All we have to do is to divide the surface into small area elements, calculate the flux at each element and add them up. Thus, the total flux f through a surface S is f ~ S E.DS (1.12) The approximation sign is put because the electric field E is taken to be constant over the small area element. This is mathematically exact only when you take the limit DS ® 0 and the sum in Eq. (1.12) is written as an integral.