Students who lived from BC to now should all know this.
A long time ago, people discovered that there are forces between charges and between magnets.
But initially, people did not connect these two functions.
Until people discovered that some stones struck by lightning were magnetic, they speculated that there might be some relationship between electricity and magnetism.
The rest of the story is very simple.
Oersted discovered that electricity can produce magnetism, and Faraday discovered that magnetism can produce electricity.
People finally realized that electricity and magnetism were inseparable, and began to use magnets to make generators and electric current to make electromagnets.
But it has been mentioned before.
Although Faraday discovered the phenomenon of electromagnetic induction and used magnet chips to represent the magnetic lines of induction.
But it was Newman and Weber who were also present in the classroom today who finally summarized the law of electromagnetic induction.
It was only in order to commemorate Faraday's contribution that they named this formula Faraday's law of electromagnetic induction.
The derivation process of Newman and Weber involves the Newman loss potential An and the Weber loss formula Aw, which is relatively complicated and will not be explained in detail here.
all in all.
The final formula of Faraday’s law of electromagnetic induction is as follows:
1.E＝nΔΦ/t
(1) When the change in magnetic flux is caused by the change in area, ΔΦ=BΔS, then E=nBΔS/t;
(2) When the change of magnetic flux is caused by the change of magnetic field, ΔΦ=ΔBS, then E=nΔBS/t;
(3) The change in magnetic flux is caused by changes in area and magnetic field. According to the definition, ΔΦ = Φ end - Φ beginning,
2. When the conductor rod cuts the magnetic field line: E=BLv
3. When the conductor rod rotates around one end to cut the magnetic field lines: E = BL2
4. When the lead frame rotates around the axis perpendicular to B: E = NBS.
Seeing these formulas, do you recall the fear of being dominated by high school physics?
Ahem
It was on this basis that Xu Yun wrote another formula that made Faraday's scalp numb:
▽×(▽×E)=▽(▽·E)-(▽·▽)E=▽(▽·E)-▽2E
▽2T=?2T/?X2+?2T/?y2+?2T/?z2.
That's right.
Smart classmates must have figured it out.
The first small formula is the triple product formula of the loss to derive the curl of the electric field E, and the second one is the Laplace of the electric field.
The name curl is also called url, which was a term proposed by Wheat in 1871.
However, related concepts have appeared as early as 1839 in the construction of optical field theory, but they have not been formally summarized.
Actually.
Based on Faraday's mathematical accumulation, he would probably not be able to understand this formula instantly and would need more in-depth analytical calculations.
However, considering that some unknown classmates almost cried when they failed the exam, let’s assume that Faraday was possessed by Gauss.
Then looking at the formula written by Xu Yun, Weber, who had the highest mathematical level among everyone present, once again realized something.
He frowned and stared at the formula for half a minute, and suddenly his eyes lit up.
Spread your left hand flat, make a fist with your right hand, and hit it hard on the palm:
“Is this the value you get from the gradient of the divergence of the electric field minus the Laplacian of the electric field?”
Xu Yun gave him a thumbs up. No wonder some people in later generations said that if Weber had not entered electromagnetism, perhaps a giant would have appeared in the history of mathematics.
This kind of thinking sensitivity is rare even in later generations.
in the above formula.
▽(▽·E) represents the gradient of the divergence of the electric field E. E(▽·▽) can be replaced by (▽·▽)E, and it can also be written as ▽2E. This leads to the following Laplacian operator
.
As long as the temperature of a point (x, y, z) in space is represented by T (x, y, z), then the temperature function T (x, y, z) is a scalar function, and the gradient ▽ can be taken on it
T.
And because the gradient is a variable, the gradient has a direction, pointing to the direction of the fastest change, so you can take the divergence ▽· for it.
As long as the expansion of the ▽ operator and the rules of loss coordinate multiplication are used, the divergence of the gradient of the temperature function T (x, y, z) can be expressed (that is, ▽2T).
Very simple and easy to understand.
Okay, the pure mathematical derivation will end here. (It has been shortened a lot. If there is any part that is difficult to understand, you can leave a message and I will try my best to answer it.)
Then Xu Yun looked at Xiaomai again and said:
"Classmate Maxwell, I will give you another task. Use the Laplacian operator to express the wave equation we obtained before."
Xiaomai's mind at this time had long been attracted by the formula written by Xu Yun. After hearing this, he almost subconsciously picked up the pen and started calculating quickly.
But I don't know why.
In his heart, he always felt that this formula was inexplicably kind.
He even had a very subtle, inexplicable feeling:
When I saw Xu Yun listed this formula.
He seemed to see his girlfriend holding someone else's hand and kissing her wantonly in front of him
Oh, I don’t have a girlfriend, so that’s okay.
And the other side.
If Xu Yun could know Xiaomai's thoughts, his face would probably look a little weird.
Because in a sense
This is indeed the behavior of a tauren:
The formula he listed is none other than one of the expressions of Maxwell's equations under the Laplace operator.
It's a pity that Xiaomai won't ask, and Xu Yun won't tell. I'm afraid this matter will become a mystery that no one knows.
Then Xiaomai took a deep breath and focused all his attention on simplifying the formula.
When Xu Yun was writing a novel in his previous life, a reader once raised a question that was quite high quality.
The one-dimensional wave equation appeared in 1746. Why do we need to re-derive the formula?
The answer is simple:
Although d'Alembert once studied the one-dimensional wave equation, what he studied was the initial solution of traveling waves.
This solution is also called the general solution, and it is actually very, very different from the wave equation of later generations.
What Xu Yun listed this time is the general solution in 1865, so there is no bug that "the wave equation has not been derived in this world line".
Don't say anything else.
Just the idea of Fourier transformation needed in the classical wave equation was not published by Fourier induction in "The Analytical Theory of Heat" until 1822.
His eyes returned to reality.
At this moment.
Like a passionate warrior of pure love, Xiaomai hummed and made calculations on the paper:
"Take the curl from both sides"
"▽·E=0"
Swish Swish Swish
With the movement of the pen tip.
The simplified data item by item appears on the paper.
As these expressions appeared, the breathing of many elders present gradually became heavier.
Except for William Whewell and Prince Albert, Wheat was the only problem solver who had not realized the seriousness of the problem.
After all, he is still just a student in the Department of Mathematics, has not yet been formally exposed to electromagnetism, and does not have enough physical sensitivity.
He only simplified and calculated the formula at the mathematical level, and at the same time did not have enough brainpower to think about the issue of 'meaning'.
However, as the calculation reaches the final stage and the answer is about to be written, no matter how slow the person is, it is time to react.
I saw this young Scottish man calculating and his pen suddenly stopped.
He raised his head in surprise and looked at Xu Yun, his face flushed:
"Mr. Luo Feng, doesn't this formula just explain"
Xu Yun nodded slightly towards him, sighed secretly, and said:
"That's right, let's finish writing it. It's time for certain things to be unsealed."
Guru
Wheat swallowed dryly and glanced quickly across the classroom.
Faraday, Thomson, Weber, Joule, Stokes
At this moment.
These elders, who occupied one-third of the thickness of later high school physics textbooks, all stared solemnly at the tip of Xiaomai's pen.
Weber's lips were trembling slightly, Faraday was holding a vial in his hand, and Stokes' fist was clenched quietly.
Even Joule's big bald head seems to reflect a lot brighter light.
They are waiting.
Waiting to witness a mathematical miracle.
"call"
Wheat's cheeks puffed up, he took a deep breath and made the final calculation on the paper.
"0 and ε0 are both constants, so the right side naturally becomes two partial derivatives of the electric field E."
"Target the negative signs again, finally"
A few minutes later.
An expression for the final term appears on the parchment:
▽2B=0ε0(?2B/?t2).
▽2E=0ε0(?2E/?t2).
The former is the equation of the electric field intensity E, and the latter is the equation of the independent magnetic induction intensity B.
As the expression was written, the classroom suddenly became audible.
Faraday gasped heavily, and once again took out nitroglycerin tremblingly and took it sublingually.
Looking at several elderly people who were as excited as Parkinson's patients, William Whewell couldn't help but look at Prince Albert and asked:
"Professors, I would like to ask you, does this expression have any meaning?"
Stokes then remembered that there were a few unknown people present, so he turned his head and explained to William Whewell:
"Mr. Whewell, you are an authority in the field of philosophy, so there may be some, uh, barriers to professional knowledge in the natural sciences."
As he spoke, he pointed to the classical wave equation derived by Xu Yun earlier and continued:
"First of all, we know that the classical wave equation derived by Luo Feng, or Mr. Feiyu, is absolutely mathematically valid."
"That is, wherever this mathematical formula is met, there must be waves."
When Xu Yun heard this, he looked at his eyes, his nose, his mouth, his mouth and his heart. He did not correct Stokes' mistake. After all, at this time, no one knew the concept of quantum.
At this time Stokes said again:
"Then Luo Feng introduced the concepts of electric field and magnetic field. After calculation, the expression still holds. What do you think this means?"
William Whewell was slightly startled, and somewhat understood what Stokes meant:
"In other words, there are waves in electromagnetic and magnetic fields?"
Faraday on the side also took a deep breath at this time, nodded heavily, and added:
"To be precise, it should be mathematically verified that electric fields and magnetic fields propagate in space in the form of waves. There is a wave that has never been discovered in the field."
"Never discovered"
Say the last thing.
Faraday's tone was almost a whisper.
Now, he finally understood what Xu Yun meant by "the seal is lifted":
There is an unknown wave in the electromagnetic field that I have studied for decades!
I didn't know anything about such an important thing before
Looking at Faraday with a gloomy expression, Xu Yun couldn't help but feel a little emotional.
When he was in high school, he once read an article by chance.
The title of the article is "Faraday's Regrets".
Of course.
This article is not a piece of chicken soup published in "Reader" or "Yilin".
Instead, it was serialized in a short essay called a study newspaper that was common when Xu Yun was studying.
That kind of newspaper costs about fifty yuan a semester, and 90% of its pages are devoted to various topics, but sometimes some articles are printed in the corners.
This kind of study newspaper and another kind of book called "Current Affairs" were the few channels through which Xu Yun could get access to social news when he was studying.
I don’t know if these things still exist twenty years later.
all in all.
In "Faraday's Regrets".
The author said that because Faraday did not receive a good education and his language skills were very low, the papers he wrote were obscure and difficult to understand.
Therefore, his series of major discoveries did not cause much shock at the time.
Wheat received an excellent education, so he summarized electromagnetic waves.
I went through a lot of articles, and finally I wrote a summary:
[Middle school and primary school are the time to learn knowledge and lay the foundation. You should learn all subjects well. Among them, Chinese class is the basic course for learning all subjects well. It is a tool class and should not be underestimated. You must not repeat Faraday's regret].
Xu Yun didn't have much idea at the time. After all, he was only in high school at that time and didn't know much about Faraday's specific life.
But when I went to college and studied the history of physics, I realized that this nonsense is nonsense.
Faraday was almost worshiped when he was alive. The generator he developed could become the soul of the second industrial revolution. How could anyone ignore him?
On the contrary, Xiaomai's career only flourished for a while while he was studying at Cambridge University, and his subsequent life has been unsatisfactory.
In addition, if we talk about obscurity, Maxwell's equations are infinitely more difficult to understand than Faraday's magnetic field lines, right?
Not to mention that Xu Yun later read the English scanned version of Faraday's paper, and the content was not difficult to understand even with 19th-century cognition.
But on the other hand.
Although Faraday himself may not have felt it until his death, from the perspective of God in future generations, electromagnetic waves can undoubtedly be said to be the biggest regret in Faraday's life.
Because of Faraday's accumulated research throughout his life, he should be able to deduce electromagnetic waves.
For example, Newman's variable potential proposed by Newman in 1845, coupled with the magnetic field law and then finding the curl, can obtain an approximation of the magnetostatic equation.
This is actually very close to electromagnetic waves.
At the same time, in some of the letters left by Faraday, later generations can also find some speculations about electromagnetic waves.
For example, in a letter to Weber in 1865, Faraday wrote:
"Perhaps there is some unknown force transmitting and interacting between the energized conductors and the conductors in the space invisible to our naked eyes."
Unfortunately, Faraday's mathematics was never good, so the person who finally predicted electromagnetic waves through derivation was Wheat, and Hertz proved it for him.
So from Xu Yun's perspective.
It is actually a pity and even unfair that Faraday did not discover electromagnetic waves.
After all, electromagnetic waves are a concept that can be called the heart of electromagnetism.
This is like a marine biologist who has studied blue whales all his life. He has an incomparable understanding of the migration routes, sounds, and living habits of blue whales.
However, due to underdeveloped deep diving technology, he had never seen a blue whale fall in his life, and he had encountered many sharks.
This is obviously a pity.
So although Xu Yun's mission target this time was wheat, after hesitating for a long time, he decided to lift the 'seal' on the electromagnetic wave.
This is also the reason why he feels indebted to Wheat and Hertz as mentioned before.
after awhile.
Faraday and others' mentality gradually returned to normal, and they began to think about other issues when they had time.
He stared at the expression derived by Xiaomai for a few seconds, frowned slightly, and said to Xu Yun:
"Classmate Luo Feng, although you have mathematically verified that there are waves in electric and magnetic fields, there are still some differences between physics and mathematics."
"If a type of substance is only established mathematically, then it can only be called a prediction at best."
"If you want to finally confirm its existence, you must come up with visible evidence and so on!"
Before finishing the second half of the sentence, Faraday suddenly realized something.
I saw his eyes fixed on Xu Yun, a hint of expectation appeared on the handsome face, and he asked:
"Classmate Luo Feng, you said before that you have two things to do today, one of which is derivation and the other is experiment."
"Could that be what that experiment is referring to?"
Xu Yun nodded gently towards him, his tone was slow but sure: