5.3: Gauss's Law

Electric fields decrease with distance from their source as \(1/r^{2}\). Compare the surface area of a cubic box with sides of length \(r\) with a sphere of radius \(r\). Their surface areas are \(6r^{2}\) and \(4\pi r^{2}\), respectively. Although the constants differ, each surface area increases by \(r^{2}\) as the size of the object increases. The observation that the field strength decreases in the same proportion as the area increases leads to Gauss's law. Nineteenth century physicists were fond of analogies between fields and fluid flow. If fluid flows from a source, then the amount of fluid flowing through any surface that encloses that source must be constant regardless of the shape of that surface. The amount of fluid passing through a surface is sometimes called the flux. Although it would be incorrect to think of electric and gravitational fields as fluids, the mathematical machinery is identical and we will need to calculate a quantity known as the electric flux that is the product of the field strength and surface area.