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It has long been discovered that there are forces between charges and between magnets, but initially people did not link these two effects. Later, it was discovered that some of the stones struck by lightning would be magnetic, and it was speculated that there might be a relationship between electricity and magnetism. Until the efforts of Oster, Faraday and others, people finally realized that the relationship between electricity and magnetism is inseparable, and people use magnets to make generators, and also use electric currents to make electromagnets.
But the deepest physical relationship between electricity and magnetism was revealed by Maxwell, who used a system of four equations to illustrate the connection between electricity and magnetism, the pair of the deepest forces in the universe, and unified the electric and magnetic fields. Maxwell's equations have been considered the most beautiful physical formula in the world since their inception.
This article will take you through the discovery process and specific meaning of Maxwell's equations, and in this process, some mathematical foundations need to be introduced. While it's hard for most people to understand this process, when you really understand Maxwell's equations, you'll be amazed as much as I am at its harmony and beauty.
< h1 class="pgc-h-arrow-right" > field and field line</h1>
In 1758, the French physicist Cullen first studied the forces between the charges and proposed Coulomb's law: the force between two charges is proportional to the product of the charge quantity, and the distance between the two is inversely proportional to the square.
Since then, scientists have been debating the way the forces act between the charges: some people think that the forces between the charges do not require time and space, and that one charge will exert a force on another charge in an instant, which is called "over-the-range action".
With the development of science, the idea of over-the-distance action is increasingly doubted. Finally, the British scientist Faraday proposed the concept of "electric field".
Faraday believed that there is a substance around the charge, which cannot be seen or touched, but it exists, and it can be transmitted in space. When an electric field is transmitted to another charge, it exerts a force on another charge. In turn, the second charge also creates an electric field, which creates a reaction force on the first charge. That is: the action between the charges is transmitted through an electric field.
In 1851, Faraday also creatively proposed a method of describing the electric field: a set of curves with arrows represents the electric field, the tangent direction of the curve indicates the direction of the electric field, and the density of the curve describes the strength of the electric field. For example, an electric field formed in space by a separate positive charge and a positive charge and a negative charge is as follows:
This method of describing the field is called the "field line", which can be used to describe the electric field and can also be used to describe the magnetic field. One can simulate field lines in a variety of ways, for example, Faraday used iron filings around magnets to simulate the situation of magnetic induction lines.
The proposal of fields and field lines provided convenience for later people to study many problems.
<h1 class="pgc-h-arrow-right" > electromagnetism and magnetism</h1>
The first person to discover the relationship between electricity and magnetism was the Danish physicist Oster.
In 1820, when Oster was giving a lesson to a student, he accidentally placed an energized straight wire above the small magnetic needle, and he was surprised to find that the small magnetic needle was actually deflected! The students present did not find the phenomenon surprising, only Auster was excited about the discovery.
After careful study, Oster proposed how the current acts on small magnetic needles. From our point of view today, Oster actually clarified that there is a circular magnetic field centered around the energized wire. Later, the scientist Ampere pointed out the direction of the magnetic field: the right-hand spiral rule.
When news of Auster's discovery that electric current could produce magnetic fields spread around the world, Faraday in England had just turned 30 and was still working under the chemist David. Many suspect david was jealous of using various methods to suppress Faraday, such as forcing Faraday to conduct optical research. It was not until David's death in 1829 that Faraday began to work on electromagnetic problems of interest to him.
Faraday believed that since electric current can generate a magnetic field, then the magnet should also be able to generate an electric current. To this end, Faraday conducted a series of physical experiments, and finally discovered the phenomenon of electromagnetic induction in 1831.
Two different wires are wound on both sides of an iron ring, and when the first wire passes through the current, the wire on the other side also generates a current. Faraday explained that this is because the current of the first circuit has changed, and the resulting magnetic field has also changed, and the changed magnetic field can produce an electric current.
We can also do such an experiment: insert a magnet into a solenoid, connect the solenoid to a ammeter, and also find that there are readings on the ammeter. This also satisfies Faraday's statement that "in the process of motion and change, the magnetic field can generate an electric current." ”
Through the discoveries of Oster, Faraday and others, it is recognized that electricity and magnetism are not separated, but closely related, and some people even think that electricity and magnetism seem to be two sides of the same problem.
<h1 class= "pgc-h-arrow-right" > the mathematical basis of Maxwell's equations</h1>
In 1860, Maxwell, a young scientist forty years younger than Faraday, came to Faraday and submitted to Faraday his previously published paper, On Faraday's Line of Force. Faraday was overjoyed and said to Maxwell: You should not limit yourself to explaining my ideas mathematically, but innovate.
Encouraged by Faraday, Maxwell further developed his own ideas and eventually summarized them into a four-equation system of Maxwell's equations. To understand these four equations, we first need two mathematical operations: fluxes and path integrals.
The first concept is flux. If the electric field E passes vertically through a plane S, we call the product of the electric field E and the area S the electric field flux. If the normals of the electric field E and plane S are sandwiched at a certain angle, we can orthogonally decompose the electric field and multiply the area perpendicular to the plane to obtain the electric field flux.
Because the electric field E can be densely represented by the electric field line, the electric field E multiplied by the area S actually represents the number of magnetic inductive wire roots that pass through this surface. If the electric fields are different everywhere, it is necessary to divide the area into infinitely many parts, adding up the electric field flux of each small part.
Expressed in mathematical terms:
Similarly, magnetic flux can be defined in the same way when a magnetic field passes through a surface. Write with integral symbols:
The second concept is path integration. If an electric field E follows the direction of path AB, multiplying the electric field E by the length L of path AB, the path integral is obtained. If the electric field E is angled to the path AB direction, the electric field is decomposed, and the field component along the AB direction is multiplied by the path length L. Magnetic fields have similar path integration.
If the electric or magnetic field is different everywhere, we can divide the path AB into infinitely more parts, and add up the path integrals of each part to represent:
Note that paths are not necessarily straight lines, and there are also path integrals along curves.
< h1 class="pgc-h-arrow-right" > Maxwell's equations</h1>
Okay, now that we know that a vector can calculate fluxes, it can also calculate path integrals. This way we can understand these four great equations.
1. The active nature of the electric field
The first equation of Maxwell's equations mathematically represents Faraday's first point of view: that charge creates an electric field in the surrounding space. The positive charge emits the electric field line outward, and the negative charge absorbs the electric field line from the surrounding area. The greater the charge, the more electric field lines are emitted or absorbed.
If we surround a charge with a closed surface, then the electric field flux on this closed surface represents the number of roots of the electric field line. Since these electric field lines are emitted by the charges within the surface, they are proportional to the algebraic sum of all the charges in the surface. It should be noted that regardless of the shape of the surface we choose, as long as it surrounds the same charge, its electrical flux is the same. If the charge is outside the closed surface, the electric field line it emits must penetrate both the surface and the surface, so that it does not contribute to the electrical flux of the surface, so the amount of charge considered in the equation is the charge inside the surface.
Write with formulas
In this formula, the left part of the equal sign represents the electrical flux on the closed surface, that is, the number of electric field wire roots that penetrate the surface, and the Σq on the right of the equal sign represents the sum of the charge algebras within the surface, and ε0 is called the vacuum dielectric constant. This equation is the first equation in Maxwell's system of equations, also known as Gauss's law for electric fields. This equation tells us that an electric field is an active field, and its source is the charge in space.
2. The passive nature of the magnetic field
Unlike electric fields, whether it is a magnetic field generated by a magnet or a magnetic field generated by an electric current, the magnetic inductance line is always closed. Magnetic induction lines have neither a starting point nor an ending point. For example, if we look at the magnetic field of the energized solenoid, we will find this feature.
Therefore, if we make a closed surface in space, the magnetic inductive line either does not penetrate the surface, or it must penetrate both the surface and the surface, so the flux of the magnetic line is zero.
In this way, the second equation of Maxwell's equations can be written:
This equation is called Gauss's law of magnetic fields, and it tells us that magnetic fields are passive, have neither a beginning nor an end, and are always closed.
3 Loop integral of the magnetic field
The third equation of Maxwell's equations is designed to explain Faraday's law of electromagnetic induction.
For example, when a magnet is close to a coil, an induced current is generated in the coil. Faraday et al. believe that this is because the magnetic flux in the coil changes when the magnet is close, and the electromotive force generated is proportional to the rate of change of the magnetic flux.
After thinking about it, Maxwell came up with the idea that the electromotive force is generated because there is an electric field force that pushes the charge, so the changing magnetic field can produce a vortex-like electric field. If a conductor happens to be in a vortex electric field, an induced current is generated in the conductor. Moreover, the magnitude of this vortex electric field is proportional to the rate of change of the magnetic flux.
So Maxwell wrote the third equation:
The left side of the equation represents the integral of the electric field path along a closed path, which can represent the electromotive force on this closed path. The right side represents the surface flux of the magnetic field change rate, that is, the rate of change of the magnetic flux.
This equation mathematically explains the origin of Faraday's law of electromagnetic induction, which can also be described as a vortex field with a vortex field.
4. Path integration of magnetic fields
Since the time of Auster, it has been recognized that there is a magnetic field around the current, and the magnetic induction intensity is proportional to the current. Maxwell wrote this feature in mathematical terms:
The left side of the equal sign represents the integral magnetic field path on an arbitrary closed path, and the right side represents the sum of the currents surrounded by this closed path.
But Maxwell's ideas were more than that. Maxwell envisioned that since a changing magnetic field can form a vortex electric field, a changing electric field can naturally form a magnetic field. For example, in a circuit there are capacitors, and during the process of charging and discharging the capacitors, there is a magnetic field around the wire. The electric field in the capacitor will change, and its status should be equivalent to the current. Thus, Maxwell proposed the concept of displacement current: a changing electric field is equivalent to an electric current.
Eventually, Maxwell wrote the fourth equation:
The left side of the equal sign represents the integration of magnetic field paths along either path, μ0 on the right represents the vacuum permeability, I represents the current, and Ф represents the electric field flux surrounded on this path. This equation states that both the current and the changing electric field can cause a magnetic field.
<h1 class= "pgc-h-arrow-right" > Maxwell's prediction</h1>
Maxwell's equations are the most beautiful physics equations ever built by mankind, and they have strong symmetry and self-consistency. It tells us that the electric and magnetic fields do not exist separately, but are unified in the electromagnetic field.
Not only that, Maxwell also calculated that if there is an oscillating electric field in the vacuum, then a magnetic field will be generated around the oscillating electric field, and this magnetic field will further generate an electric field... In this reciprocation, the electromagnetic field can propagate to a distance, forming electromagnetic waves.
Maxwell calculated the speed of electromagnetic waves, found that the speed of electromagnetic waves in a vacuum is exactly equal to the speed of light, and boldly predicted that light is an electromagnetic wave.
At this point, classical physics has reached the extreme, and the confidence of physicists has been extremely inflated. So much so that at the gathering of physicists in 1900, Sir Kelvin proudly declared that the edifice of physics had been basically completed, and that future generations would only need to do some more tinkering.
Unfortunately, Maxwell did not personally confirm the electromagnetic waves it predicted, and in 1879, maxwell died at the age of 48. And in the same year, Einstein, the greatest scientist of modern physics, was just born.
Before Maxwell, the greatest physicist was Newton, because his law of universal gravitation unified the heavens and the earth, and he proved that the moon and the apple met the same laws of physics. The greatest physicist after Maxwell was Einstein, because his special and general relativity unified time and space, giving people the idea that the world actually exists in a unified space-time. Between Newton and Einstein, the greatest physicist was Maxwell, whose system of equations unified the electric and magnetic fields, and the electromagnetic waves he predicted became the most important means of communication in modern times. Even Einstein's theory of relativity is partly to deal with the covariance of Maxwell's equations.
Although some people's life expectancy is not ordinary, their brilliant ideas remain in the world forever.
Maxwell
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