Essentially, the entire theory of electromagnetism can be found in the following four equations:

\begin{aligned}\mathbf{\nabla \cdot E} &= \frac{\rho}{\epsilon_{0}} \\ \mathbf{\nabla \times E} &= - \frac{\partial{\mathbf{B}}}{\partial{t}}\\ \mathbf{\nabla \cdot B} &= 0\phantom{\frac{1}{2}}\\ \mathbf{\nabla \times B} &= \mu_{0} \mathbf{j}+\mu_{0} \epsilon_{0} \frac{\partial{\mathbf{E}}}{\partial{t}} \end{aligned}

These are Maxwell’s Equations in differential form, not in integral form, which is the way they are often introduced. I will discuss them in this form, however, as I believe the differential equations convey more elegantly their physical meaning straight from the mathematics. Let’s get started.

### Fields

If you ever did high school physics, you should have some idea of what electric and magnetic fields are. Below is an example of each (depicted using field lines):

What these field lines show is the direction of the respective fields at each location (indicated by arrows) as well as their relative strengths (indicated by density of field lines). So what are these fields actually representing? Well, for electric fields, this shows you the direction an object with a positive electric charge would be forced, while a negatively charged object would feel a force in the opposite direction. As for magnetic fields, you could think of it as similar to electric fields, except now it’s north poles of magnets that are forced along the field lines, while south poles are forced in the opposite direction.

Maxwell’s Equations will tell us about the patterns these fields make, how the fields arise, and how they can affect each another.

### Divergence

Ok, so a little bit of vector calculus is needed, but it’s not hard, I promise. The divergence of a field is quite self explanatory: it tells you how much a field diverges from a region. If the field is pointing mostly away from a region, we say the divergence is positive; if it points towards, divergence is negative. The divergence is zero if neither is the case (or if it is a perfect balance of the two). Here are examples of positive and negative divergence, respectively:
In maths-lingo, the divergence of the field $\mathbf{F}$ is written as $\mathbf{\nabla \cdot F}$, where the upside-down triangle thingy is called the del operator. Notice that two of Maxwell’s Equations use divergence, one for the electric field and one for the magnetic field.

### Curl

This one is also quite self explanatory: the curl of a field tells you how curly it is in a particular region, i.e. how loopy it is. Here are two examples of curly fields:

The curl of a field $\mathbf{F}$ is written as $\mathbf{\nabla \times F}$. You may notice that this is a vector quantity, unlike divergence, because of the cross product. To find the direction of this vector, simply curl (excuse the pun) your right hand fingers in the direction of the curly field, and your thumb will be pointing in the direction of the curl vector. So for the above diagram, the anti-clockwise curling field will have a curl vector point out the page (towards your face).

Hurricanes/typhoons/cyclones/willy-willies (depending on your nationality) are great examples of both divergence and curl. Air circulates clockwise/anti-clockwise around these weather systems (depending on your hemisphere) and thus the air velocity displays a curly pattern like above. In fact, it also displays negative divergence, since the air also converges to the centre of the storm.

### Maxwell’s Equations – Statics

We’re fully equipped to start dissecting these. We’ll start with electrostatics and magnetostatics (or together, statics). This is the case when the electric and magnetic fields we’re studying are constant (not changing with time). What this allows us to do is ignore any time derivatives of fields in Maxwell’s Equations, so we’re left with something somewhat simpler:

\begin{aligned}\mathbf{\nabla \cdot E} &= \frac{\rho}{\epsilon_{0}} \\ \mathbf{\nabla \times E} &= \mathbf{0}\\ \mathbf{\nabla \cdot B} &= 0\phantom{\frac{1}{2}}\\ \mathbf{\nabla \times B} &= \mu_{0} \mathbf{j} \end{aligned}

An explanation of symbols. We dealt with divergence and curl earlier, so you should easily spot those. $\mathbf{E}$ and $\mathbf{B}$ are the symbols used for electric and magnetic fields, respectively. The symbols $\epsilon_{0}$ and $\mu_{0}$ are just constants (important constants, but we’ll ignore them for now). $\rho$ is called the electric charge density, i.e. how much charge we have in a region. Similarly, $\mathbf{j}$ is the current density, i.e. how much current is passing through a particular area.

Gauss’s Law: $\displaystyle\mathbf{\nabla \cdot E}= \frac{\rho}{\epsilon_{0}}$
After that lengthy preamble, this equation should be easy to read: the divergence of an electric field is proportional to the charge density. The first important thing this tells us is that electric fields have divergence, i.e. they display patterns similar to ones seen earlier when describing divergent fields (see diagrams). Secondly, the amount of divergence is determined by the amount and type of charge we’re dealing with. Positive charges result in positive divergence of the electric field, and vice versa. Also, the more negative/positive charge we have in a region, the more strongly the surrounding electric field will point towards/away from that region.

Faraday’s Law (incomplete): $\displaystyle\mathbf{\nabla \times E} = \mathbf{0}$
Electric fields have no curl (under static conditions). That may sound kinda boring, but it’s important: we can’t have an electric field that displays a curly pattern (see diagrams). From Gauss’s Law we know that electric fields point away from positive charges and towards negative ones. If that is the only way electric fields are created, then it should be clear that constructing a circular electric field is impossible.

Gauss’s Law for Magnetism: $\displaystyle\mathbf{\nabla \cdot B}= 0\phantom{\frac{1}{2}}$
Magnetic fields have no divergence. In other words, we’ll never see magnetic field lines ever starting or ending, like we do when looking at field lines entering/emanating from electric charges. When magnetic field lines hit the south pole, they go right through the magnet and out the north pole. This tells us that we can’t isolate a north or south pole. If you try and break a magnet into two, another north/south pair will simply emerge and you’ll have two magnets.

Ampere-Maxwell Law (incomplete): $\displaystyle\mathbf{\nabla \times B}= \mu_{0} \mathbf{j}\phantom{\frac{1}{2}}$
The curl of a magnetic field is proportional to the current density (under static conditions). From this we see that magnetic fields are inherently curly! And that curl is proportional the amount/density of current passing through that area. This suggests that if we have a current-carrying wire, it can induce a curly magnetic field (notice how the right hand rule can be used to relate the direction of the field/current):

You may be asking: “But what about bar magnets? I see no current, but there’s a magnetic field!” This is because there is current, just on the microscopic scale, down to electron orbitals and spins. These individual moving charges cause their own curly magnetic fields, and these gazillion contributions add up to one big magnetic field that you can feel on a macroscopic scale, even though there isn’t a macroscopic current.

### Maxwell’s Equations – Dynamics

We can learn a lot from Maxwell’s Equations under static conditions, but the story isn’t complete without touching on the non-static cases. Let’s now deal with the two equations containing derivatives:

Faraday’s Law: $\displaystyle\mathbf{\nabla \times E}= - \frac{\partial{\mathbf{B}}}{\partial{t}}$
Now we find that electric fields can have curl, but only proportional to the rate of change of the magnetic field. So if the magnetic field isn’t changing (static) then we won’t find curl in the electric field (just like before), but if it is changing, we will, and the faster it changes, the more curly electric field we will get. This gives us our first link between electric and magnetic fields. We can even use this link to explain electromagnetic induction, i.e. the reason why moving a magnet in a coil creates electricity. This is because the changing magnetic field induces an electric field which curls with the coil, causing electric charges to flow around the coil, and thus we have electricity. Next time you think of things you’re grateful for, remember Faraday’s Law.

Ampere-Maxwell Law: $\displaystyle\mathbf{\nabla \times B}= \mu_{0} \mathbf{j}+\mu_{0} \epsilon_{0} \frac{\partial{\mathbf{E}}}{\partial{t}}$
We see now that not only can our old curly magnetic field come from electric currents, but it can also come from changing electric fields! This is another link between magnetic and electric fields. There is much more to this equation than meets the eye, but unfortunately I can’t show where these results come from without us having a bit of extra knowledge, so I’ll quickly go on a physics rant about what this law can do: the equation forms the basis of a wave equation, providing the mathematical description for electromagnetic radiation, i.e. propagating electric and magnetic fields. Infrared, radio waves, UV and light are all forms of electromagnetic radiation. If it isn’t enough for you that this equation (with some help from the others) tells you why you can even see things, then it also predicts that these waves will have a constant speed of ${1}/{\sqrt{\mu_{0} \epsilon_{0}}}$. This so happens to be equal to the speed of light. I told you $\mu_{0}$ and $\epsilon_{0}$ were be important!

### Summary

After all of that, it’s easy to summarise what we now know about electric and magnetic fields. Electric fields tend to diverge/converge towards a region, depending on the type and amount of charge in that region. Electric fields can also display curly patterns, but these can only be induced by the presence of a changing magnetic field. Magnetic fields never diverge/converge towards a region. Rather, magnetic fields are curly in nature and can be created in two ways: from a current passing through a particular area, and/or from a changing electric field in that area. It is from these laws that pretty much all of electromagnetic phenomena can be explained.

Image sources:
Electric Field, Magnetic Field, Magnetic Field of Current

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