Using refrigerator magnets and a bubble bath to understand Maxwell’s first law.

“I have also a paper afloat, with an electromagnetic theory of light, which till I am convinced to the contrary, I hold to be great guns”-James Clerk Maxwell

Electromagnetism, in fact, is a great gun. The inventions of the near past ranging from microwave ovens to mobile phones to bullet trains, everything is based on the principles of electromagnetism. Each of these everyday go-to devices packs some really complex mechanisms inside of them, transistors, semiconductors, transformers and it goes on. But despite the incredible complexity, these functions could be worked out almost completely with just four simple sets of equations put forth by James Clarke Maxwell, which brilliantly explain electricity and magnetism, or collectively electromagnetism.

The first equation, although very simple, might be pretty intimidating at first. So let us break it into parts. The “B” here expresses the magnetic field at a point in space, oh but wait what’s a magnetic field?

Let’s get this out of the way first. Have you ever used a refrigerator magnet? Have you ever noticed how it quickly slips out of your grip and sticks to the metal surface of the refrigerator, when brought close to it? Well, that can’t happen on its own and neither is someone or something forcing it to stick to the surface, (unless you live in a haunted house) so there must be something around in the air that’s causing the magnet in the refrigerator magnet to get attracted to the refrigerator. This “something” is what we call the magnetic field, in other words it is the region around a magnetic substance, the refrigerator magnet, where it can exert a force on another magnetic object, the refrigerator surface. This magnetic field is what determines the force that will act, so the stronger the magnetic field at a place the stronger the force. If you were to be much more critical about this little experiment, you must also have noticed that it always sticks in a particular direction, it is the plane surface that always sticks to the refrigerator and not the transverse, as in the curved surface, so this magnetic field is not completely constant around the magnet. There are two particular places that tend to get attracted more, these two places are what we call the poles (north and the south) of the magnet, or in other words, the places that have the strongest magnetic field around them. As the magnetic force varies according to a particular side of a magnet we must have something to express this. We use something called the magnetic field lines to express this force, the magnetic field lines for the refrigerator magnet would look something like this.

If you look closer, the poles, where the force is maximum, Is where the magnetic field lines are the densest, in other words, the closer the magnetic field lines are packed and the lesser the angle to which they are tilted to each other, stronger the force. This is where the “B” comes in, inMaxwell’s equation. It is a mathematical equation that expresses the strength of the magnetic field along with the direction in which it acts, I.e it is a vector!. This simple equation has large implications in the ways it helps us decipher the direction along with the strength of the force that would act on an object kept close to a magnet.

Now that we have understood this part of the equation, let’s move on to understand what the inverted triangular thing represents. It is what we call divergence. Understanding divergence gets much easier when we correlate it with fluid flow. Consider you are in your bathtub because that’s the best place for doing physics after all. We are going to consider a slightly modified version of a bathtub to makes things easier to understand, the tap and the sink in the bathtub look something like this. (Here the tap and sink are not on the same side to make understanding things easier)

Say you’ve left the tap open for the bathtub to fill up and also leave the sink open, I know it’s bad to waste water but it’s all for the sake of science, and it’s a thought experiment anyway so it does no harm. The water flow as seen from above must look something like this. The little arrows drawn all around represent the flow of water, it is important to mention that we are neglecting any reflections from the side surfaces of the bathtub, but if we were to look at the actual flow in a more holistic manner the water comes in from the tap end and flows out of the sink. The arrows at a particular point direct towards the flow of water at that point, so there can be an array of velocities in the tub. So if we were to consider the flow of water in, say, the yellow sphere in the middle of the tub, the amount of water that flows in also flows out, so the Divergence of the water-flow around this yellow sphere would be zero, which means there is, overall, no flow in or out of the sphere.

Now consider the maroon sphere around the tap end, the flow just goes outwards or it acts as a source of the water flow or does it? The flow that we perceive as going outwards is actually being supplied by the tap above, the water is still coming from above and leaving from below and what ever water comes from the tap, flows out too! So, basically the magnetic field here is also zero. Okay! so now lets see if its the same with the sink, what ever water comes in from the tub flows out at the same rate from the sink as well! So the net flow here is zero too! Now you may ask, “Well, if the divergence is always zero whats the point of having it anyway?”

Actually the divergence is not always zero for other cases, the most prominent one being electric fieldlets first consider a very simple case: a light bulb! Consider a similar imaginary sphere around the bulb now, you can tell that the light emitted from the speaker just moves directly out of the sphere, this is a case for positive divergence, here the divergence of the light emitted around the sphere is positive! But is there negative divergence as well? Yes, a vacuum cleaner! Again consider an imaginary sphere all around the vacuum cleaner, it pulls the dust particles in itself, and hence when we compute the overall divergence of the dust around the vacuum cleaner, it turns out to be negative!

Now that we have a vague notion of what “B” and “the downwards pointing triangle” are, we can begin to understand Maxwell’s first equation. The equation says that the divergence, or the net flow, of the magnetic field, is always zero no matter what the position considered or the magnetic substance chosen is. Let us recall the experiment we did with the bathtub and get back the field lines in the refrigerator magnet to get a better view.

Consider the orange-red sphere on the right side of the magnet, you see the net magnetic field that flows in, flows out as well, but does this still hold for EVERY point? Let’s see the circle near the pole of the bar magnet, it at first looks like it is a source.

Though this might seem like a “source,” but it isn’t, we have actually not shown the magnetic field lines that flow in from the South Pole to the North Pole inside the refrigerator magnet. So if we were to follow the actual magnetic field lines, as shown above considered overall space, you see the net flow still sums up to zero, for all points around the magnet. If you pay close attention you may have noticed how similar the bath tub and the magnet are when compared with respect to the divergence of their respective fields. Thus, this equation suggests that the net flow of the magnetic field at any point in space is zero, no matter where the magnet or the point is!

This has some really serious implications. This means that there are no individual sources or sinks for the magnetic field. Or in other words, a magnet always has one north and one South Pole, which you may have noticed if you have ever broken a piece of magnet. The two individual broken magnets, still attract from one end and repel from another.

However many researches have been conducted, and many including the London Centre for Nanotechnology have found evidence of the existence of magnetic monopoles but none, to date has been confirmed. But if one is ever found, it would mean that Maxwell’s equations are wrong and so is our understanding of electromagnetism. This would be a new era in the field of Physics!

Hello there! I am Ahitagni and I love math, physics & cheese!

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