Now that we have added this final addition to Ampere's Law, we have a set of four equations that fully describe the sources of electric and magnetic fields from charges. Together, these four equations are called Maxwell's Equations and help us describe everything from how electric and magnetic fields are created to how light travels.

First we have Gauss's Law, which says that charges make electric fields: $$\int \vec{E} \bullet d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$$ These electric fields point away from positive charges and towards negative charges.

Next we have Faraday's Law, which says that a changing magnetic field makes a curly electric field: $$-\int \vec{E} \bullet d\vec{l} = \frac{d\Phi_B}{dt}$$ These electric fields point in a direction that oppose the change in flux that created them.

After that we have Gauss's Law for magnetic fields. We did not end up spending much time with this equation since it's result is rather simple: $$\int \vec{B}\bullet d\vec{A} = 0$$ This equation says that there is no magnetic field that points away or towards a source. This tells us that there are no magnetic monopoles (i.e., a north pole cannot exist without a south pole).

Finally, we have Ampere's Law, which says that a current (moving charges) OR a changing electric field can make a curly magnetic field: $$\int \vec{B} \bullet d\vec{l} = \mu_0 I_{enc} + \mu_0\epsilon_0\frac{d\Phi_E}{dt}$$

These equations are important because, as we learned, once you have the electric or magnetic field, you can relate those fields to the electric or magnetic force; to energy, electric potential, or work; and apply those principles to circuit applications like capacitors, resistors, and current. Ultimately, with the conservation of charge, these Maxwell's equations govern how charged particles behave and interact.