Johann Karl Friedrich Gauss and the Gauss’s Law

José A. Rivera Ríos

Karl Friedrich Gauss, (30 April 1777 – 23 February 1855) born in Braunschweig Germany, was scientist and a mathematician who contributed considerably too many fields. Gauss had an amazing influence in many areas of science and mathematics and is ranked as one of history's most important mathematicians. Gauss was a child prodigy a person that from at an early age masters one or more skills at an adult level. There are many anecdotes pertaining to his precocity while a child and he made his first ground-breaking mathematical discoveries while still an adolescent. Gauss was a phenomenal mental calculator. He completed Disquisitiones Arithmeticae, his masterpiece, in 1798 at the age of 21, though it would not be published until 1801. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the current day. He developed an important relation, no know as Gauss’s law, it is a mathematical statement of the relation between the amount of charge enclosed in a region of space and the total flux of the electric field over the surface enclosing that region of space. The total electric flux depends on the total charge enclosed and not how that charge is distributed. Understanding Gauss's law requires understanding the concepts of electric fields and electric flux. Electric charges exert forces on each other without physical contact because each charge causes an electric field and the electric fields exert forces on other charges. Electric fields are the intermediary causing electric charges to exert forces on each other.

The electric flux through an area is defined as the electric field multiplied by the area of the surface projected in a plane vertical to the field. It is an important tool since it permits the assessment of the amount of enclosed charge by mapping the field on a surface outside the charge distribution. For geometries of sufficient symmetry, it simplifies the calculation of the electric field. If the electric field is not perpendicular, then this product must also be multiplied by the cosine of the angle between the electric field direction and the outward pointing line perpendicular to the surface. This line perpendicular to the surface is called the normal. In terms of vector multiplication, this product is the scalar or dot product of the electric field vector and the area vector. Gauss' law is a powerful tool for the calculation of electric fields when they originate from charge distributions of sufficient symmetry to apply it. If the charge distribution lacks sufficient symmetry for the application of Gauss's law, then the field must be found by summing the point charge fields of individual charge elements. It also offers a simple way to determine the electric field when the charge distribution is simple and possesses a high degree of regularity. In 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism, and the discovery of Kirchhoff's circuit laws in electricity. In my opinion Gauss is one of the greatest mathematics and scientist on the history of humanity and his inheritance will remain in world thru the ends of times.