Let There Be… Electricity? Maxwell’s Equations: Explained
This is the fourth and presumably the last part in the Maxwell’s equations series. I’ll try to keep this one short.
If you haven’t already, I recommend checking out the other three parts:
The fourth Maxwell equation, known as Ampere’s Circuital Law, is what we’ll be working with here. It looks, in its differential form, something like this:
What we first see is a quantity known as the curl of the magnetic field, B. This is denoted by the downward triangle and the B with the little arrow on top.
The curl at a point in a field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation.
We’ll get back to this in a moment.
If you’ve read any of my previous articles, I trust that you’re familiar with the other quantities. For the uninitiated, here’s what they are:
Mu naught is the magnetic constant, also known as the permeability of free space. Essentially, it’s the magnetic version of epsilon naught. It determines how magnetic fields can exist within empty space for magnets of given strengths. Mu naught value is the measure of the amount of resistance experienced against the formation of the magnetic field in a vacuum.
Next, “J” is actually Maxwell’s addition to an equation that already existed but wasn’t quite accurate enough. It’s a quantity known as displacement current. Basically, it represents the current passing through any surface around the region of space we are studying. But we can break things down further.
Ordinary electric currents, called conduction currents, whether steady or varying, produce an accompanying magnetic field in the vicinity of the current. Maxwell, in the 19th century, predicted that a magnetic field also must be associated with a changing electric field even in the absence of a conduction current, a theory that was subsequently verified experimentally. As magnetic fields had long been associated with currents, the predicted magnetic field also was thought of as stemming from another kind of current. Maxwell gave it the name displacement current, which was proportional to the rate of change of the electric field that kept cropping up naturally in his theoretical formulations.
Of course, the displacement current is not a current at all. It is, in fact, associated with the generation of magnetic fields by time-varying electric fields. Maxwell came up with this rather curious name because many of his ideas regarding electric and magnetic fields were completely wrong. The inclusion of the displacement current in these equations treat electric and magnetic fields on an equal footing: i.e., electric fields can induce magnetic fields, and magnetic fields can induce electric fields.
The epsilon naught, here, is the electrical equivalent of mu naught. It represents the permittivity of free space. It is the property of every material, which measures the opposition offered against the formation of an electric field. It tells the number of charges required to generate one unit of electric flux in the given medium. In other words, permittivity is a representation of how many charges are required to allow one unit of electric field lines to pass through one unit of area. In really hand-wavy terms, it can be thought of as a measure of how much a vacuum can be polarised by an electric field.
This part, finally, is a quantity that represents the rate of change of the electric field. The weird ds tell us we’re dealing with partial derivatives. For now, however, we don’t need to necessarily bother about what that’s about. We know that this is physically synonymous with the rate at which the electric field changes. In other words, it’s the change in the electric field per unit time.
Finally, back to the curl. From a mathematical perspective, we’d be looking at something like this:
Physically, however, that daunting expression doesn’t help us much. We can do better. Curl, briefly explained, is a quantity that measures the tendency of rotation around a point. Think of it like dropping a twig in a bathtub. If the twig merely floats, then you’d have zero curl. If it rotates clockwise, negative curl. Anticlockwise? Positive curl. The quantity is quite literally what it says. The curl, like the divergence, associates each point in 2D space with a number, rather than a vector. Although the intuition for these kinds of quantities is given in terms of fluid flow, they’re both incredibly significant in the face of other fields too.
Now, I trust that you understand what the curl is. It’s a quantity that measures the circulation of a vector field. In other words, it’s a measure of the tendency of a vector field to “swirl around”.
Now, putting it all together without delving into its nuances, this equation tells us that a magnetic field can be generated by a changing electric field or a flow of charged particles into or out of a region of space. In essence, it completely encapsulates the relationship between electric and magnetic fields.
Mathematically, however, what you’d understand is this: the curl of the magnetic field at some point is equal to the magnetic constant multiplied by the sum of the displacement current and the product of the permittivity of free space and the rate of change of the electric field at that point.
Needless to say, that sounds a lot worse. This is why physics stays and math only comes along. Anyway, I’d suggest you focus more on the physical interpretation. Anyway, this concludes the Maxwell series. I hope you now have an understanding of what any of this really means. If I’ve made an error, feel free to tear me apart in the comments. And as always, thank you for reading.
