Gauss' Law

Carl Friedrich Gauss was a brilliant mathematician, physicist and astronomer. His mathematical works included many theories and papers in the realm of pure math, such as Number Theory. In the physics world, he is most remembered for his law regarding electric fields.

All Electric charge exact forces upon each other, which is regulated by Coulomb's Law

F = k * q1 * q2 / r * r, where constant k = 8.9*10^9).

Electric fields make everything simpler by graphically showing the strength of attraction or repulsion by one particular charge (it doesn't emit a field, it is imaginary). The strength of the field is based on the strength of the charge. The field would then exert a force upon any charge in its vincinity (technically infinite, but at very long distances it is practically zero). On a positive test charge, the strength of the force would be

F = E * q

Where E is the strength of the field in newtons per coulomb and q is charge of the test charge.

This all builds up to the concept of electric flux. Flux is the a measurement of the amount of electric field passing through an area. The formula is given by:

Flux = (Integral) E dA, A is area, E is field strength

What has it to do with Gauss then? Well he made some obsevations regarding flux of enclosed surfaces (also known as Gaussian surfaces). For an charge enclosed in a space, no matter how you move the space around, warp it, stretch it, or whatever, as long as the charge is still inside, the flux remains constant. Here is an analogy. You get a box and put a positive charge inside. If you increase the dimensions of the box by a factor of 2, then at the stretched surface of the Gaussian space, the stregth of the field is decreased. However, the area of which the weakened field is going through increases by the same factor. Gauss came out with three conclusions:

Whether there is a net outward or inward electric flux through a closed surface depends on the sign of the enclosed charge

Charges outside the surface do not giv a net electric flux through the surface

The net electric flux is directly proportional to the net amount of charge enclosed within the surface but is otherwise independent of the size of the closed surface

Simplified, Gauss' Law is an alternative way to express the relationship between electric charge and electric field (the other way is Coulomb's Law). It states:

The total electric flux through any closed surface (a surface enclosing a definite volume) is proportional to the total net electric charge inside the surface.

The equation is:

Flux = (Intergral) E dA = Qenclosed / e , where Q is the algebraic sum of the charge inside the enclosed space, e is 8.854 X 10^-12

Hence electric flux passing through a closed surface is really just dependent on charge.

Gauss' Law is not just an alternative way to state the relationship between electric charge and electric field (the other being Coulomb's Law). In fact, Coloumb's Law can be derived directly from Gauss' Law--the two are truly the same law, stated in different terms. Note: in my notation, variables in bold are vector quantities. ε₀ is the permitivity of free space and is 8.85 x 10^-12 Farads/meter or coloumbs-squared/Newtons-meters-squared. All integrals are to be understood as loop integrals, over the entire Gaussian surface.

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Take a point charge +q, around which is envisioned a Gaussian concentric spherical surface of radius r. Divide this surface into differential areas dA. By definition, the area vector for each area dA is dA with magnitude equal to the area and direction perpendicular to the surface (directed outward from the interior of the sphere). From the symmetry of the sphere, we also know that at any point on the sphere, the electric field vector E is also perpendicular to dA and directed outward, and so is parallel to dA. Therefore the angle between E and dA is zero.

Gauss' Law is usually stated:

ε₀∫E•dA = q_enclosed

Since E and dA are parallel, their dot product is simply EdA. The enclosed charge is simply q (since we made our sphere around this charge), and is a constant. E is also a constant, because it varies only radially for a point charge, and we have restricted the radius to the radius r of our sphere. So Gauss' Law can be rewritten:

ε₀E∫dA = q

The loop integral of the differntial areas is just the area of the sphere, 4πr². Substituting, we have:

ε₀E(4πr²) = q.

This is:

E = q/(4πε₀r²)

which is Coloumb's Law! :-)

Btw, if anyone knows how to make loop integrals in HTML, I'd love to know!