The purpose of this paper is to examine and explain Gauss's Law, a fundamental concept in the study of electricity. Gauss's Law is an expansion of Coulomb's law in that it provides an alternative, and sometimes much easier, method of calculating electric fields. I will begin this exploration by introducing the concept of electric flux, and then define Gauss's Law, giving two examples of its application. Next, I will explain the properties of an electrostatic conductor in equilibrium using Gauss's Law, and conclude with a brief comparison to Ampere's Law.
2 Electric Flux
We know that the strength of an electric field is determined by the number of electric field lines per area. The number of field lines that penetrate a given area is
proportional to what is called, an electric flux. In the figure to the right,
we see electric field lines passing through a surface of area A. Let - = AÌ‚ , the area
vector, which has a magnitude of the surface A, and points in the normal direction Ì‚ If the
surface is placed in a uniform electric field - that points in the same direction as Ì‚, the flux through the surface is:
- - - and if - makes an angle θ with Ì‚, then the
equation becomes:
- -
.
.
A surface can also be curved and - can also vary across the area. Suppose, in a closed surface, we divide the area into a larg number of small
elements called (as shown). The electric flux through
- -
-- , then is:
-
Ì‚
Ì‚
The total flux of the entire surface can be found by summing all area elements. As the number of elements approaches infinity, the sum becomes an integral, and we have the general equation for electric flux:
Replacing 4π
()
with yields: (Gauss's Law)
3
Gauss's Law
∫- -
FFrom this equation, we can make the statement that the net flux through a closed surface is proportional to the net charge enclosed. This is known as Gauss's Law. Even though we have chosen a sphere in this example, Gauss's Law holds true regardless of the shape of the surface. This can be proven...
(Will expand)
-- proof w/ diagrams --
Now that we have defined electric flux of a closed surface, we can describe its relationship to an enclosed charge. This surface is called a Gaussian surface. Suppose a point charge Q is enclosed by a sphere with radius r.
From the formula derived in the previous section, we know that the net flux through the Gaussian surface is:
∫- -
Since the electric field is constant and the surface area of a sphere is 4 , we can define its flux as:
4 Applications of Gauss's Law
Gauss's Law is especially useful when finding the electric fields of very symmetric charge distributions.
1. As our first example, let us consider a very long cylinder of uniform charge density. We know that the charge distribution of a very long rod is cylindrically symmetrical.
Therefore, since the electric field points outward symmetrically in all directions, we can choose our Gaussian
surface to be a coaxial cylinder of length l and radius r. The amount of charge enclosed by the Gaussian surface is .
Hence the flux through the Gaussian surface is:
∫- - ∫- - ∫- - = 0 + 0 + = E( )
Applying Gauss's law gives:
E( )= Solving for E gives:
This is the same result obtained using Coulomb's law. Notice that the electric field depends only on the inverse of the radius and is independent of the length.
2. Gauss's Law can also be applied to an infinite plane of charge, with uniform charge density σ. Since the plate has planar
symmetry, - at all points must be constant point perpendicular to the plane.
The through our cylinder goes through three surfaces: s1, s2, and s3.
Since - is perpendicular to the surface at both ends of the cylinder (s1 and s2), no flux passes
through there.
flux
We choose a cylinder to be our Gaussian surface. Again, the flux can go through three surfaces in the cylinder: s1, s2, and s3.
- is parallel to the curved part of the cylinder (s3), so the flux is zero there. Therefore, the flux is:
∫- - ∫- - ∫- - = + +0 = ( )
Since the two ends of the cylinder are the same distance from the plane, E = , so the equation becomes:
Applying Gauss's Law gives:
Solving for E gives:
From the final equation we see that E does not depend on the distance from the plane, so the field is uniform everywhere in an infinite plane of charge.
5 Conductors
Conductors are materials where the electrons inside move around freely. All conductors share several basic properties.
The electric field inside a conductor is always zero. For example, in the figure below, a solid conducting sphere is placed in an external electric field. As soon as the
external field (- ) is applied, electrons in the conductor rush towards the left side of the sphere, leaving a
positive layer of charge on the right. These regions of charge create an electric field of
their own (- ). The charges will continue to accumulate on both sides until the internal field cancels the external field. At this point, electrostatic equilibrium, the net field inside the conductor will be zero.
Any net charge a conductor carries resides on its surface. This can be explained using
the previous property and Gauss's Law. Suppose we make our Gaussian surface inside the conductor, and arbitrarily close to
the surface. Since we know that - is
zero everywhere inside a conductor,
the only component that remains is the flux through end outside the conductor:
∫- -
where σ is the surface charge density at that point.
6 Ampere's Law
Ampere's law in magnetism is analogous to Gauss's law in electrostatics. Just as Gauss's law gave us a method for solving for the electric field in a given charge distribution, Ampere's law gives us a method for solving for the magnetic field in a given current configuration. It simply states that the line
integral of -- - is proportional to the current enclosed by the loop, I.
∮ -- - (Ampere's Law)
Like Gauss's Law, Ampere's Law also requires a high degree of symmetry in order to apply. Ampere's law can also be applicable to current configurations such as a long cylinder or an infinitely large sheet. However, instead of using the Gaussian
then - must be zero Gaussian surface. As a result, the net flux through the Gaussian surface is also zero. Therefore, we can conclude that any net charge resides on the conductor's surface.
Furthermore, we know that - is normal to the surface of a conductor, otherwise electrons on the surface would experience a net force and move around. (The conductor would not be in equilibrium in such a case.) To find the strength of the electric field just outside of the conductor, we can use Gauss's law, and choose a cylinder to be our Gaussian surface. As noted earlier, there is no flux through the curved part of the cylinder because it is parallel
with - . The end of the cylinder inside the conductor also has zero flux because there is no charge inside. Thus
everywhere on the
surface, Ampere's law uses an Amperian loop, a closed path of any shape surrounding a current.
7 Additional Aspects
-Gauss's law for magnetism -Gauss's law for gravity -Faraday's Cage (will expand)
8 Conclusion
As a conclusion, I will sum up some of the most important concepts that we have discussed.
Electric flux is the product of the magnitude of the electric field E and surface area A. It is proportional to the number of electric field lines going through that surface.
A Gaussian surface is a closed surface through which the flux of an electric field, due to an enclosed charge, is calculated. Rearranging the equation for a charge enclosed by a spherical surface, we get the
important result: . This equation, known as Gauss's law, can be applied to a
number of different cases where charges are symmetrically distributed. We applied this to a symmetrical cylinder and a large plane.
Another important use of Gauss's Law is its role in describing the properties of conductors in electrostatic equilibrium. All conductors in equilibrium exhibits certain traits, notably that the electric field inside is them is always zero, any net charge resides on its surface, and the vector E is always perpendicular to the surface of the conductor.
Gauss's Law also shares some similarities with Ampere's law, a law relating magnetic field and current. Like Gauss's Law, Ampere's Law requires a high degree of symmetry in order to use. Unlike Gauss's Law, however, it uses a loop (Amperian loop) as its path of integration, rather than a surface.
So as a whole, Gauss's law provides us with a relatively simple way to solve complicated problems using qualitative reasoning. It is a fundamental law in the study of electrostatics; and without it, many of the electromagnetic phenomena that exist today would become very hard to explain.
