Maxwell's equations are the Holy Grail of electricity and magnetism. They are to classical electromagnetism what Newton's laws are to classical mechanics. The complete set of Maxwell's equations is displayed below.
Gauss' law for electricity states that the electric flux through a closed surface is proportional to the charge the surface encloses. Mathematically, this is represented as:
This equation is a simple way to determine the electric field. To determine an electric field, we select a Gaussian surface through which the electric field is constant at all points. This allows us to pull the \(\mathbf{E}\) out of the integral.
Gauss' law for magnetism states that the magnetic flux through a closed surface is always zero. In effect, this means that no magnetic monopoles exist (that is, you cannot separate a north and a south pole). Shown as:
Faraday's law tells us that a changing magnetic field creates an electric field. In fact, this law is not new, and is a simple restatement of something you have already studied earlier. You were probably taught Faraday's law as \(\varepsilon=-\frac{d\Phi_B}{dt}\). We arrive to the new statement by the simple substitution \(\varepsilon=\oint \mathbf{E}\cdot d\mathbf{l}\). Thus,
The Ampere-Maxwell law tells us essentially the opposite of Faraday's law is true: changing electric fields creates a magnetic field.