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 θ with it, the projected area in a plane perpendicular to v is dS cos θ. Therefore, the flux going out of the surface dS is v.n̂ dS. For the case of the electric field, we define an analogous quantity and call it electric flux.

Electric flux is a concept adapted from the idea of fluid dynamics. When liquid flows through an area, we can determine how much liquid is flowing by looking at the area and the flow rate. In a similar fashion, electric flux quantifies how much electric field passes through a given area. Imagine placing a flat surface in an electric field. The amount of the electric field lines crossing that surface represents the flux. When the surface is perpendicular to the field, the maximum amount of electric field lines pass through. However, if the surface is tilted, fewer lines cross due to the angle, leading to a projection factor (cos θ). This concept allows us to calculate 'how much' electric field is interacting with a certain area, which is crucial for understanding how electric fields behave.

Think of electric flux like the sunlight entering through a window. On a sunny day, if you have a window perfectly facing the sun, you would get maximum sunlight streaming into the room, just like maximum electric field lines would cross a surface normal to their direction. If you tilt the window, less sunlight enters, similar to how electric field lines would pass through at an angle. By learning to adjust the window, or in this case, the area, we can manage how much light (or electric flux) we want in the room.

In the picture of electric field lines described above, we saw that the number of field lines crossing a unit area, placed normal to the field at a point is a measure of the strength of electric field at that point. This means that if 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 θ. Clearly, the number of field lines crossing the area element will be smaller. The projection of the area element normal to E is DS cosθ. Thus, the number of field lines crossing DS is proportional to E DS cosθ.

The electric flux (Φ_E) through an area is mathematically defined using the equation Φ_E = E · DS = E DS cos θ. Here, E is the magnitude of the electric field, DS is the area vector representing the surface through which the field lines pass, and θ is the angle between E and DS. When the surface is oriented such that θ is 0 (perfectly aligned), all the electric field lines pass through, and thus the flux is maximum. When θ is 90 degrees (surface parallel to field), no field lines pass through, so the flux is zero. This relationship allows us to quantify the electric flux through any surface based on its orientation concerning the electric field.

Imagine holding a fishing net (the area) in a flowing river (the electric field). If you hold it perpendicular to the flow, you catch the most fish (maximum flux). If you tilt the net so it is at an angle, you catch fewer fish because the flow is less effective at going through the net (the projection decreases). If you lay the net flat on the water (parallel), you catch no fish at all.

The total flux through a surface S is given by the integral Φ_E = ∫S E · D*S. 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 through a closed surface is f ~ S E.DS.

To find the total electric flux through a surface, we can visualize the surface as composed of many tiny flat areas (elements). For each tiny area, we calculate its contribution to the electric flux using the aforementioned relationship and then sum up all of these contributions. This summation can be expressed as an integral, leading to a more general formula for calculating electric flux when dealing with continuous surfaces. This concept is crucial in practical applications, like computing the electric flux around complex shapes and configurations.

Think of this process like measuring rainfall over a large area using small rain gauges. Each gauge collects the amount of rain (flux) over a small area. By summing all the readings from each gauge across the larger area, you can determine the total amount of rain that has fallen. Similarly, integrating the electric flux over small areas of a surface gives you the total flux through that surface.

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