October 6: Gauss's Law In Electrostatics
Meaning of Equation:
The electric field follows the inverse square law and the source of electric field can be a single electric charge.
There are two important things which this equation tells us. The first thing is that the electric field follows an inverse square law, just like the gravitational field. This fact isn't apparent in the equation of Gauss's law but you know what - Gauss law holds true for electric field because of this inverse square property. Also, why the R.H.S of this equation is what it is? Let us learn.
We'll start off with something familiar. Look at the image above. This is the illustration of an inverse square law. Did you notice something ? As we go farther from the light source, the area is getting larger. Suppose we take our surfaces at 1 m, 2 m, 3 m and 4 m as shown. The area and the quantity of light (intensity) falling on the respective surfaces is also given. If you multiply area with light quantity for any surface, you always get 1. The reason for this is the conservation of light energy.
Now let us just assume that such a conservation holds for the electric field as well. Then the electric field must be following the inverse square law just as in the example above. Replace the light source by an electric charge and the corresponding light rays by the electric field of lines and imagine that the setup forms a closed volume bounded by surfaces 1 and 4. What do you observe now? It is now clear that the field lines are entering our imaginary volume from surface 1 and leaving it from surface 4. The strength of the field lines is decreasing by 1/r^2 and the area is increasing by r^2. So the product is independent of r, where r is the length of side. Thus, the net electric flux through this volume is zero as the same amount of field lines are entering and leaving this particular volume.
Mathematically, we may write ∫ E.da = 0 where the integral on L.H.S represents the quantity known as the electric flux. Take any random surface and you can break it into such hypothetical cones and prove that the net flux through any surface is zero. Notice that this is true only as long as the charge q is outside a particular volume.
As soon as the charge goes inside the volume, you can still make pairs on opposite sides of the bounded surfaces but this time the sign (lines leaving or entering the surface) of electric flux will be same. This means there will be net flow of electric field lines from inside the surface. So let us assume that there is a charge q inside an irregular surface S as shown in the figure below. We want to find the net electric flux through this surface. It won't be zero as the charge is enclosed inside the volume V. We'll use a trick.
The trick is to draw an imaginary spherical surface S' of radius r. The electric field everywhere on its surface is (q/4 πεo r^2) and is directed normal to the surface. Now the total flux through this surface will be the product of electric field and the total area of surface. The area is 4πr^2. So the net flux through S' is q/εo. Now the flux outward through S, the bigger surface, will also be q/εo as the flux is conserved.
That's it, we just derived the Gauss law in electrostatics, that the "net flux through any closed surface is q/εo, where q is the charge enclosed by the surface". Did you see how beautiful this result is? Had the electric field not been an inverse square field, we would never have anything like Gauss's law in electrostatics. Thus the Gauss's law guarantees that the field we are talking about, follows the inverse square law, that the intensity of that field dies out by a factor of the square of the distance from the source!
The equation given above is nothing but the differential form of what we just derived. The second meaning of this equation follows directly. The divergence of electric field is non zero. This means that even if you place a single electric charge in a closed surface, it will have its own field lines that won't end anywhere, unlike a magnet, whose field lines start from one end and end at the other. Thus single electric charges can exist in space. They need not be in pairs, like the poles of a magnet.