1.13. GAUSS’S LAW

As a simple application of the notion of electric flux, let us consider the total flux through a sphere of radius r, which encloses a point charge q at its centre. Divide the sphere into small area elements, as shown in Fig. 1.22.

Gauss's Law describes the relationship between electric flux and charge. Specifically, it states how much electric field (flux) penetrates a closed surface surrounding a charge. When we have a sphere enclosing a point charge, the law helps us calculate the total flux through the surface of the sphere.

Imagine a balloon filled with air. When you inflate the balloon, the air inside exerts pressure uniformly on the inside surface of the balloon. Similarly, electric field lines distribute themselves uniformly over the surface of the sphere, creating constant electric flux as long as the charge remains at the center.

The flux through an area element DS is q\nD f= EDi =S 4p e r2 rˆiD S (1.28)\n0\nwhere we have used Coulomb’s law for the electric field due to a single charge q. The unit vector rˆ is along the radius vector from the centre to the area element. Now, since the normal to a sphere at every point is along the radius vector at that point, the area element DS and rˆ have the same direction.

In this part, we derive the total electric flux through the sphere by integrating the electric field across the surface area. Since the electric field due to a point charge is radial, the flux through each area element of the sphere can be calculated as the product of the electric field strength and the area element.

Think of the sun's rays entering through a circular window. The intensity of light hitting any part of the window depends on how many rays hit that area, much like the electric field hits the area element of the sphere.

The total flux through the sphere is obtained by adding up flux through all the different area elements: q\nf = S D =S S 4p e r2 allD S 4pe r2\n0 0\nSince each area element of the sphere is at the same distance r from the charge, S = S = S 1 2.

By taking into account all the area elements on the surface of the sphere, we find the total flux. Because of symmetry, every area element is subject to the same electric field strength, making the calculation straightforward. We multiply the average flux over a single area element by the total surface area of the sphere to find the total flux.

Consider measuring rainfall with a round bucket. Regardless of where the bucket is placed, as long as it is exposed to the rain, you can calculate the total volume collected by understanding how much rain falls on each area of the bucket's top surface.

Equation (1.30) is a simple illustration of a general result of electrostatics called Gauss’s law. We state Gauss’s law without proof: Electric flux through a closed surface S = q/e (1.31) 0 q = total charge enclosed by S.

Gauss’s law simplifies the calculation of electric fields by relating the electric flux through a closed surface to the charge enclosed by that surface. This fundamental principle holds for any shape of a closed surface, emphasizing the power of symmetries in electrostatics.

Imagine a net that catches fish (the closed surface) in a lake where you have a specific number of fish (the charge). No matter how you position the net, the number of fish caught by it directly relates to how many fish are present in that portion of the lake.

The law implies that the total electric flux through a closed surface is zero if no charge is enclosed by the surface.

When applying Gauss's law in cases where there are no charges inside the closed surface, it logically follows that the outward flux must balance itself, leading to a net flux of zero. This has practical implications for understanding electric fields in various configurations of charge.

Think of a sealed jar that you are observing from outside. If there's nothing inside the jar, no matter how you look at it (changing angles, perspectives), you can't see anything—just as you cannot have net electric flux through a surface enclosing no charge.

--