Section 22.4 and 23.1 in Matter and Interactions (4th edition)

## Putting Together Maxwell's Equations

Now that we have added the 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.

### The Four Maxwell's Equations

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.

### Limitations on Classical E&M

This theory of electromagnetism is classical in that it applies to systems of many atoms and electrons. We have constructed arguments for single charges or even small atomic systems, but in reality, these charges and systems are governed by quantum mechanics. So, while we might have a classical picture of the atom and electron cloud, a better model is more complicated and requires a quantum theory of electromagnetism in which both atomic systems and the field are treated quantum mechanically. (These topics are typically introduced in a modern physics course.) This reformulation of electromagnetism helps us do cutting research and development as electronic systems become increasingly smaller and push on the limits of quantum mechanics.

Even so, this classical theory is one of the most complete theories in science for the range of physical systems it can describe. It works at the astronomical scale and is consistent with relativity, and it works down to the microscopic level (up to the quantum limit). It is truly an incredible theory.