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Understanding Gauss's Law

Electric Field, Flux, Closed Gaussian Surface, and Enclosed Charge

Jun 12, 2008

The total electric field integrated over a surface enclosing a volume of space gives the total amount of electric charge in that region of space.

Gauss's law is a mathematical relationship 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 Field and Electric Flux.

Like electric charges repel and unlike charges attract. 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.

Physicists measure the total amount of electric field acting over some surface area by the electric flux.

If the electric field direction is perpendicular to the surface, then the formula for electric flux is the electric field, E, multiplied by the surface area, A. 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, as shown in the equation in the figure.

If the electric field is not constant, it is necessary to use calculus techniques to integrate EdA over the total surface area.

Gauss's Law

If we consider any arbitrarily shaped volume of space, Gauss's law relates the total charge enclosed within that volume to the total electric flux over the surface of the region. This surface is often called a Gaussian surface in honor of Karl Gauss, originator of Gauss's law.

Gauss's law states that the total electric flux over a closed surface equals the total charge enclosed by that surface divided by a constant. It is given in equation form in the figure. In a vacuum this constant is called the permittivity of free space and is denoted by the Greek letter, epsilon, with a 0 subscript.

Calculations with Gauss's Law

Gauss's law can provide a useful tool to calculate electric fields from various charge distributions. However if the electric field is not constant calculating the electric flux requires doing a difficult integration. In this case Gauss's law is valid, but it is not a very useful computational aid. Gauss's law is useful in problems having some type of symmetry making the electric field constant over the Gaussian surface. Then a difficult integration is unnecessary, and a seemingly difficult computation becomes less difficult. The key to making Gauss's law a useful for calculations is taking advantage of symmetries in the problem.

Consequences of Gauss's Law

Physics textbooks often derive Gauss's law using Coulomb's law for the force between electric charges. However if we consider Gauss's law as a more fundamental law than Coulomb's law, then the fact that electrical forces are inverse square forces can be considered as a consequence of Gauss's law.

The fact that the electric field is zero inside a conducting shell also derives from Gauss's law and is the principle behind the Faraday cage.

Gauss's law is one of the fundamental principles of electromagnetic theory and is important enough to have become one of Maxwell's equations for electromagnetic theory.