Witness the moment of miracle: How to deduce electromagnetic waves from Maxwell's equations?

　　In the previous two articles, Nagao introduced the integral and differential forms of Maxwell's equations. As we all know, Maxwell derived electromagnetic waves from this set of equations, and then found through calculation that the speed of electromagnetic waves is exactly equal to the speed of light. Therefore, Maxwell predicted that "light is an electromagnetic wave", and this prediction was later confirmed by Hertz.

　　The discovery of electromagnetic waves made Maxwell and his electromagnetic theory a god, and also brought human society into the radio age. You can now call your friends at any time and read long-tail technology articles on your mobile phone, which are closely related to electromagnetic waves. So, how did Maxwell derive the electromagnetic wave equation from the Maxwell equations? In this article, let's witness this miraculous moment together.

　　01What is a wave?

　　To understand electromagnetic waves, we must first understand what waves are. Some people may find this question a bit strange. Is it necessary to ask what waves are? If I throw a stone into the water, a wave will form on the water surface; if I shake a rope, a wave will appear on the rope. There are many such wave phenomena in life. Although I have little education, I still know what waves are.

　　Yes, water waves and ripples on a rope are all waves. I am not asking you to count on your fingers which things are waves and which are not. I am asking you: What are the common characteristics of all these things called waves? How can we describe waves using a unified mathematical language?

　　When we study physics, we summarize a certain consistency from the myriad of natural phenomena, and then use mathematical language to quantitatively and accurately describe this consistent phenomenon. Now we have discovered that many phenomena such as water waves and waves on a rope have such a wave phenomenon, so we naturally have to look for the unified mathematical law behind this wave phenomenon, that is, to find the equation that describes the wave phenomenon, that is, the wave equation.

　　In order to find a unified wave equation, let us first look at the simplest wave: if we shake a rope, a wave will appear on the rope and move along the rope. If we shake it at a constant frequency, a continuous wave will appear.

　　In order to better study the waves on the rope, we first establish a coordinate system and then focus on one of the waves. Then, we see a wave moving in the positive direction of the x-axis (to the right) at a certain speed v, as shown below:

　　So, how do we describe this fluctuation?

　　First of all, we know that a wave is constantly moving. The picture above is just the wave at a certain moment. It will move a little to the right at the next moment. It is also easy to calculate how much it has moved: because the wave speed is v, after Δt time, the wave will move to the right by a distance of v·Δt.

　　In addition, I don’t care what shape the wave is at this moment. I can think of it as a collection of points (x, y), so that we can describe it with a function y=f(x) (a function is a kind of corresponding (mapping) relationship. In the function y=f(x), for every given x, a y can be obtained through a certain operation f(x). This pair of (x, y) forms a point in the coordinate system, and connecting all such points gives us a curve).

　　Then, y=f(x) only describes the shape of the wave at a certain moment. If we want to describe a complete dynamic wave, we have to take time t into account. That is to say, our waveform changes with time, that is: the vertical coordinate y of a point on my rope is not only related to the horizontal axis x, but also to time t. In this case, we have to use a binary function y=f(x,t) to describe a wave.

　　This step is easy to understand, it simply tells us that waves change with time (t) and space (x). But this is not enough, there are things in the world that change with time and space, such as apples falling and basketballs flying in the sky, what is the essential difference between them and waves?

　　02The Nature of Waves

　　If we think about it carefully, we will find that when waves propagate, although the location of the waves is different at different times, their shapes are always the same. That is to say, the wave was in this shape one second ago, and one second later, although the wave is no longer in this place, it still has this shape. This is a very strong restriction. With this restriction, we can distinguish waves from other things that change in time and space.

　　Let's think about it this way: since we use f(x, t) to describe the wave, the initial shape of the wave (the shape at t=0) can be expressed as f(x, 0). After time t, the wave speed is v, so the wave moves to the right by a distance of vt, that is, the initial shape f(x, 0) is moved to the right by vt, so the result can be expressed as: f(x-vt, 0).

　　Why is it that when a function graph is shifted to the right by a certain amount vt, the result is the function's independent variable x minus vt, rather than plus vt? This is a middle school math problem, and I'll briefly review it for you: Imagine that if I shift the graph of a function f(x) to the right by 3, then the original value f(1) at 1 will now be the function value at 4. So, if you still want to use the function f(x), you must subtract 3 from 4 (to get the value of f(1)), rather than adding 3 (4+3=7, f(7) doesn't make any sense here).

　　Therefore, if we use f(x, t) to describe the wave, then the wave at the initial moment (t=0) can be expressed as f(x, 0). The image of the wave after time t is equal to the image at the initial moment shifted to the right by vt, that is, f(x-vt, 0). Therefore, we can mathematically give the essence of wave motion:

　　In other words, as long as there is a function that satisfies f(x,t)=f(x-vt,0), and the shape at any time is equal to the initial shape translated by a certain distance, then it represents a wave. This is true for water waves, sound waves, waves on a rope, electromagnetic waves, and gravitational waves, which is also consistent with our intuitive understanding of waves.

　　Here we have given a description of the wave from a purely mathematical perspective. Next, let us analyze the reasons for the formation of the wave from a physical perspective to see if we can obtain more information.

　　03Tension

　　When a rope is placed on the ground, it is still. When we swing it, a wave will appear. Let's think about it: how does this wave spread to a distant place? Our hand only pulls one end of the rope and does not touch the middle of the rope, but when the wave reaches the middle, the rope does move. The movement of the rope means that there is a force acting on it (this is what Niu Jue taught us). So where does this force come from?

　　With a little analysis, we will find that this force can only come from the interaction between adjacent points on the rope. Each point "pulls" the point next to it, and the point next to it moves (just like when we line up and count, we only notify the person next to you). This force between the ropes is called tension.

　　The concept of tension is also easy to understand. For example, if we pull a rope with force, I clearly exert a force on the rope, but why doesn't the rope get stretched? Why doesn't the point closest to my hand get pulled?

　　The answer is naturally that the points near this point exert an opposite tension on this particle, so that this point is pulled by me on one side and by its neighboring points on the other side, and the effects of the two forces cancel each other out. However, the force is mutual. The nearby point exerts a tension on the endpoint, so this nearby point will also be subject to a pulling force from the endpoint. However, this nearby point does not move, so it will inevitably be subject to the tension of the inner point. This process can continue to propagate, and the final result is that all parts of the rope will be tense.

　　Moreover, we can also conclude that: if the mass of the rope is negligible, the rope is not knotted and is not stretched, then the tension inside the rope is equal everywhere (as long as there is a point where the tension on both sides is unequal, then this point should be pulled away and the rope will be deformed). This is a very important conclusion.

　　Through the above analysis, we know that when an ideal rope is in a taut state, there is equal tension inside the rope. When a rope is stationary on the ground, it is in a relaxed state and has no tension, but when a wave reaches here, the rope will become a wave shape, and then there is tension. It is this tension that makes the points on the rope vibrate up and down, so analyzing the influence of this tension on the rope becomes the key to analyzing wave phenomena.

　　04 Wave force analysis

　　Then, we will select a small section AB from the rope in a wave state, and analyze how this small section of rope moves under the action of tension. Don't worry, we will not involve any complicated physical formulas here. The only formula we need is the famous Newton's second law: F=ma.

　　Newton's first law tells us that "an object will remain stationary or in uniform linear motion when there is no force acting on it or the net external force is zero." So what if the net external force is not zero? Newton's second law goes on to say: If the net external force F is not zero, then the object will have an acceleration a, and the relationship between them is quantitatively described by F=ma (m is the mass of the object). In other words, if we know the mass m of an object, as long as you can analyze the net external force F it is subjected to, then we can calculate its acceleration a based on Newton's second law F=ma, and knowing the acceleration will tell us how it will move next.

　　Newton's second law combines the force (F) and motion (a) of an object. If we want to know how an object moves, we just need to analyze what forces it is subject to. That's why it's awesome.