Gauss' law (electrostatics)

In physics, more specifically in electromagnetism, Gauss' law is a theorem concerning an integral over a closed surface of an electric field. In vacuum Gauss' law takes the form (in SI units):

$\iint _{\mathrm {closed} \atop \mathrm {surface} }\mathbf {E} \cdot d\mathbf {S} ={\frac {Q_{\mathrm {tot} }}{\epsilon _{0}}}.$

Here dS is an vector with length dS, the area of an infinitesimal surface element on the closed surface, and direction perpendicular to the surface element dS, pointing outward. The vector E is the electric field at the position dS, the dot indicates a dot product between the vectors E and dS. The double integral is over a closed surface that envelops a total electric charge Q_tot. The constant ε₀ is the electric constant. The law is called after the German mathematician Carl Friedrich Gauss.

Application to spherical symmetric charge distribution

Gauss' law is a convenient way to computing electric fields in the case of spherical-symmetric charge distributions. For instance, a point charge is a spherical-symmetric charge distribution. Another example is a charged, conducting, spherical shell, the charge distribution being homogeneously distributed over the shell.

Take the origin of a spherical polar coordinate system in the center of symmetry of the charge distribution—the position of the point charge, or the center of the spherical shell, respectively. Because of symmetry, E has a radial component only (parallel to the unit vector e_r). Moreover, this component does not depend on the polar angles,

$\mathbf {E} =E_{r}(r),\quad E_{\theta }=E_{\phi }=0.$

Take a sphere of radius r as the closed-surface to integrate over (in case we are considering a spherical shell r is larger than the radius of the spherical shell); the surface element is

$d\mathbf {S} =r^{2}\,\sin \theta \,d\theta \,d\phi \,\mathbf {e} _{r}\quad {\hbox{and}}\quad \mathbf {E} \cdot d\mathbf {S} =E_{r}(r)r^{2}\sin \theta d\theta d\phi$

Then,

${\frac {Q_{\mathrm {tot} }}{\epsilon _{0}}}=E_{r}(r)\,r^{2}\iint _{\mathrm {surface} \atop \mathrm {of\,sphere} }\sin \theta \,d\theta \,d\phi \quad \Rightarrow \quad E_{r}(r)={\frac {Q_{\mathrm {tot} }}{4\pi r^{2}\epsilon _{0}}},$

where r is the distance of the field point to the origin. In the case of a point charge we have proved here Coulomb's law from Gauss' law. In the case of a charged spherical shell, we find that the electric field is such that it seems that the total charge on the shell is concentrated in the center of the shell and that Coulomb's law applies to the charge concentrated in the center.

Literature

C. F. Gauss, Allgemeine Lehrsätze in Beziehung auf die im verkehrtem Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs und Abstossungskräfte [General theorems regarding the attractive and repulsive forces that act with inverse ratios of the square of the distance]. Carl Friedrich Gauss, Werke, Gesellschaft der Wissenschaften zu Göttingen, Göttingen (1877) Vol. 5, pp. 195-242. This work was first published in: Resultate aus den Beobachtungen des magnetischen Vereins im Jahre 1839 [Results from the observations of the magnetic Society in the year 1839], pp. 1-51.