Application of Gauss's Law

Gauss's law is particularly useful in computing or where the charge distribution has some symmetry. We shall illustrate the application of Gauss's Law with some examples.

1.An infinite line charge

As the first example of illustration of use of Gauss's law, let consider the problem of determination of the electric field produced by an infinite line charge of density r_LC/m. Let us consider a line charge positioned along the z-axis as shown in Fig. 2.4(a) (next slide). Since the line charge is assumed to be infinitely long, the electric field will be of the form as shown in Fig. 2.4(b) (next slide).

If we consider a close cylindrical surface as shown in Fig. 2.4(a), using Gauss's theorm we can write,

.....................................(2.15)

Considering the fact that the unit normal vector to areas S₁ and S₃ are perpendicular to the electric field, the surface integrals for the top and bottom surfaces evaluates to zero. Hence we can write,