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

　　08Classical Wave Equation

　　Let's look at μ/T on the right side of the equation. If you carefully calculate the unit of μ/T, you will find that it is exactly the reciprocal of the square of the speed. That is to say, if we define a quantity as the square root of T/μ, then the unit of this quantity is exactly the unit of speed. As you can imagine, this speed is naturally the propagation speed v of this wave:

　　With the velocity v defined in this way, our final wave equation can be presented:

　　This equation is the classical wave equation we are looking for. Why is it called the classical wave equation? Because it does not take into account quantum effects. In physics, classical is synonymous with non-quantum. If we want to consider quantum effects, this classical wave equation is useless. We must use the quantum wave equation instead, which is the famous Schrödinger equation.

　　Schrödinger started from this classical wave equation and combined it with de Broglie's concept of matter waves to guess the Schrödinger equation. This equation freed physicists from the fear of being dominated by Heisenberg's matrix and returned to the beautiful world of differential equations. Although the Schrödinger equation is powerful, it does not take into account the effects of special relativity. Particles moving at high speeds (near the speed of light) are very common in the microscopic world. We also know that when an object approaches the speed of light, the relativistic effects must be considered, but the Schrödinger equation does not do this.

　　It was Paul Dirac who finally made the Schrödinger equation relativistic. Dirac locked himself in a room for three months and finally came up with the equally famous Dirac equation. The Dirac equation theoretically predicted antimatter (positrons) for the first time. Although scientists at the time thought that Dirac was kidding, Chinese physicist Zhao Zhongyao observed the annihilation of positrons and electrons in the laboratory for the first time at almost the same time.

　　In addition, Dirac's work also promoted the birth of quantum field theory, opening the door to a fascinating new world. Physicists followed this path to tame electromagnetic force, strong force, and weak force, and established the standard model of particle physics. As a result, the world was peaceful and stable, except for that damn gravity. We will talk about these wonderful stories later. If these stories are written into a book "Quantum Heroes", well, it must be no less than Jin Yong's martial arts.

　　OK, back to the point. Seeing that this classical wave equation can still stir up such a big wave, do you suddenly have awe for it? We have derived the classical wave equation through such operations. Some of you may be a little confused. It doesn’t matter. Let’s sort it out again. This seemingly complicated equation containing second-order partial derivatives actually just tells us: if we regard this tiny section of the rope as a particle, then this particle satisfies Newton’s second law F=ma, that’s all.

　　09 Review

　　Our entire derivation process is nothing more than looking for these three quantities in F=ma. We decomposed the tension of the rope in the vertical direction, and then obtained its resultant force F (T·sin(θ+Δθ)-T·sinθ) in the vertical direction; we defined the mass per unit length μ, and then we can calculate the mass m (μ·Δx) of the small rope; through the analysis of the wave function f(x,t), we found that if we take a partial derivative of this function representing distance (displacement) with respect to time, we get the velocity, and take another partial derivative to get the acceleration, so we get the acceleration a of this rope (??f/ ?t?). Then we put these quantities together according to Newton's second law F=ma.

　　In the process of solving the problem, we made a lot of approximations: because we have a very small section, we can use Δx to approximate the length of the rope Δl; assuming that the disturbance is very small and the rope deviates very little from the x-axis, then the angle θ is very small, so we use the tangent value tanθ to approximate the sine value sinθ. At first glance, many people wonder how such a strict derivation can be so casually approximated? If you approximate here and there, will the final result still be accurate?

　　To understand this problem, you have to learn calculus formally. Now I tell you that the core idea of ​​calculus is to approximate a curve with a straight line. Do you believe it? In calculus, various small segments of straight lines are used to approximate curves, but the results are very accurate. Because we can make these line segments very, very small, or infinitesimal, then the error will gradually become infinitesimal. So when we analyze this rope, we also emphasize that we take a very small segment, give a very small disturbance, and get a very small angle θ.

　　In addition, tanθ is a first derivative, and then their difference is divided by Δx once, and another first derivative appears, so the left side of the equation has two partial derivatives of f(x, t) with respect to position x. The right side of the equation is the acceleration a obtained by taking two partial derivatives of the function f(x, t) with respect to time t (taking the first derivative gives the speed, and taking it twice gives the acceleration).

　　So, although what we see is a wave equation, it is actually just a disguised version of Newton's second law F=ma. With this understanding, the wave equation is not strange at all. Let's take a closer look at this equation:

　　The meaning of this wave equation is also very intuitive. It tells us that a function f(x,t) that changes with time t and space x, if the ??f/ ?x? obtained by taking the derivative twice with respect to space x and the ??f/ ?t? obtained by taking the derivative twice with respect to time t satisfy the above relationship, then f(x,t) describes a wave.

　　If we solve this equation, we get the function f(x, t) that describes the wave. When we did mathematical analysis on the wave earlier, we came to the following conclusion: if a function f(x, t) describes a wave, then it must satisfy f(x, t) = f(x-vt, 0). Therefore, the solution f(x, t) of the wave equation must also satisfy the above relationship. Friends who are interested can prove this by themselves.

　　Well, let's stop here for the classic wave equation. With the wave equation, you will find that we can derive the equation of electromagnetic waves from Maxwell's equations through a few simple calculations, and then determine the speed of electromagnetic waves.

　　10 Maxwell's equations in a vacuum

　　The differential form of Maxwell's equations is this:

　　The origin and development of this set of equations has been introduced in detail in the previous article "The Most Beautiful Formula: Maxwell's Equations You Can Understand (Differential Calculus)" by Changwei Technology, so I will not elaborate on it here. In this set of equations, E represents the electric field intensity, B represents the magnetic induction intensity, ρ represents the charge density, J represents the current density, ε0 ​​and μ0 represent the dielectric constant and magnetic permeability in vacuum (both are constants), ▽ is the vector differential operator, ▽· and ▽× represent the divergence and curl respectively:

　　Our next task is to see how to derive the equations for electromagnetic waves from this set of equations.

　　First, if a wave can really be formed, then this wave must propagate outwards. It can also propagate itself in a place far away from charges and currents (that is, where there are no charges and currents). Therefore, we first let the charge density ρ and the current density J both equal 0. When ρ=0 and J=0, we get the Maxwell equations in a vacuum:

　　Some people wonder how you can make the charge density ρ equal to 0? Then the first equation becomes the divergence of the electric field ▽·E=0, which means the electric field strength E is equal to 0, and there is no electric field. How can there be electromagnetic waves without an electric field?

　　Many beginners have this misunderstanding: they think that if the divergence of the electric field ▽·E equals 0, then there is no electric field. In fact, the divergence of the electric field equals 0, which only tells you that the electric flux through the infinitesimal surface containing this point is 0. The electric flux of 0 does not mean that the electric field E is 0, because the electric flux (the number of electric field lines) entering and leaving this surface is equal. In this way, there are as many negative electric fluxes (the number of electric field lines coming out) as there are positive electric fluxes (the number of electric field lines entering), and the positive and negative electric fluxes in and out cancel each other out, so the total electric flux is still 0. Therefore, the divergence ▽·E of this point can be 0, but the electric field strength E is not 0.

　　So everyone must make a clear distinction: the divergence of the electric field E is 0 does not mean that the electric field E is 0, it just requires the electric flux to be 0, and the same goes for the magnetic field.

　　Let's look at Maxwell's equations in vacuum (ρ=0, J=0): Equations 1 and 2 tell us that the divergence of the electric and magnetic fields in vacuum is 0, and equations 3 and 4 tell us that the curl of the electric and magnetic fields is equal to the rate of change of the magnetic and electric fields. The first two equations describe electricity and magnetism independently, while the last two equations describe the relationship between electricity and magnetism. We can also vaguely feel that if you want to derive the equations for electromagnetic waves, you must combine the above equations, because waves need to propagate outward, and the individual equations above only describe the curl or divergence of a certain point.

　　There is a very simple way to combine them all: take the curl on both sides of Equation 3 and Equation 4 again.