I just want to be a quiet scholar

Chapter 86 Estimation of Gauss's Theorem

"This classmate, please stay awake and don't doze off in the exam room." The invigilator reminded Shen Qi very seriously.

"Well, I'm sorry." Shen Qi rubbed his temples. Recently, he revised the manuscript of "Mathematics Skills of Mathematical Olympiad Champion Shen Qi", which made him feel haggard and didn't rest well.

"Hurry up and finish the last calculation problem, and go home to make up for sleep." Shen Qi cheered up and carefully reviewed the problem.

After reviewing the last 25-point calculation question, Shen Qi finally became somewhat interested: "Just this last question, it's like a question from a regular physics teacher."

Most of the physics calculation problems are equipped with schematic diagrams. Physics problems without diagrams generally show two extremes, one is simple to doze off, and the other is difficult to hang up.

In the preliminary round, the town house question has the highest score, 25 points. This calculation is accompanied by a schematic diagram.

The schematic diagram is a circle, and a dotted line R is drawn from the center O to the seven o'clock position of the circumference, and this dotted line R is the radius of the circle.There is a small black dot mq not far from the center O.

The text description of this question is:

"As shown in the figure, a uniformly charged ring with a linear charge density of λ (λ>0) and a radius of R is fixed on a smooth horizontal insulating table. A smooth ball with mass m and charge q rests on the table. It is very close to point O in the center of the ring."

"Assuming that the distribution of electric charge on the ring is not affected by the electric charge of the ball, try to judge whether the movement of the ball afterward is vibration?"

"If it is vibration, set the distance r0 <<R between the initial position of the ball and point O, and try to estimate the vibration period T of the ball by an appropriate approximation method."

The difference between estimation and rigorous calculation is that estimation can bypass complex mathematical calculations and directly obtain correct qualitative conclusions and relatively rough quantitative results.

As for the last calculation question in the preliminary round, whether the motion of the ball is vibrating or non-vibrating, Shen Qi must give a qualitative conclusion, and the judgment must not be wrong.This is the first step.

For the same physics problem, if the estimation method is used, there are often more than one options.

Obviously, this is a question of electromagnetism. Shen Qi chose the Gauss theorem of electrostatic field as the basis to start answering the question among many estimation methods.

Shen Qi made an auxiliary diagram, taking the axis passing through point O and perpendicular to the plane of the ring as the x-axis.

On the circle plane, take point O as the center, and draw a circle of radius r.

Extend this circle by l in the positive and negative directions of the x-axis, and a cylindrical surface is formed.

Shen Qi took this cylindrical surface as a Gaussian surface, because there is no charge in it, according to Gauss's theorem, it can be obtained:
E*ds=0
As soon as Gauss's theorem came out, the truth became clearer.

The electrostatic force on the positively charged ball always points to the O point in the center of the ring, which is a restorative conservative force. The movement of the ball is vibration, and the vibration center is the O point.

Shen Qi quickly solved the first question, which is to give a qualitative conclusion, and students who have received training in physical competition should be able to give a correct conclusive judgment.

The second question asked Shen Qi to estimate the vibration period T of the ball, which was a little more troublesome.

The electric flux at both ends of the cylinder can be calculated approximately by the electric field strength on the x-axis. Shen Qi made the calculation:
E1=λ（2πR）l/4πε（R^2+l^2）^3/2=λRl/2ε（R^2+l^2）^3/2
Then the approximate value of the electric flux through the two ends comes out:
Both ends E*ds≈E1*2πr^2
The electric flux on the side of the cylinder can be approximated by the electric field intensity Er at a distance r from point O on the circular plane. According to Gauss's theorem, it can be obtained:

Cylindrical surface E*ds=End surface E*ds+Side surface E*ds=0
Then the electrostatic force on the charged ball at r is:
Fr=qEr=-λq/4εR^2*r
Considering the linear restoring force, the small ball will make simple harmonic vibration around the point O under its action.

So period T=4πR root sign εm/λq
"It's done." After being baptized by the CMO and even the IMO, Shen Qi is already a seasoned veteran in the discipline competition.

Whether it is digital competition or material competition, the competition mode is similar.

Since he is a veteran, he cannot be arrogant, irritable and exaggerated, and must always maintain a rigorous style of competition.

Shen Qi checked the test paper and handed it in. At this time, more than 30 minutes had passed since the test started.

"This Nima?"

"Fuck?"

"Submission so early?"

"Write non-stop, it takes 30 minutes to fill the test paper, right?"

"I don't need to think about this stuff? Or do I just write it blindly?"

The other contestants in the same examination room were obviously affected by Shen Qi's handing in, and each shouted MMP in their hearts.

Although the exam papers for the CPhO National Preliminary Competition are not difficult, it is rare to submit papers so early.

There are also sharp-eyed players in the competition circle: "This product is submitted in advance... Could it be Shen Qi? The champion of the International Mathematical Olympiad?"

Shen Qi walked to the entrance of the examination room, Ah Ni, suddenly sneezed, and a chill appeared on the spine.

"Who is scolding me?" Shen Qi turned his head suddenly, and glanced sharply at the players.

"Grass!"

The players chrysanthemum trembled, and they continued to answer the questions.

"Hey, please leave the exam room quickly after handing in the paper." The invigilator urged Shen Qi and quickly checked Shen Qi's exam paper. It was filled with everything that should be written. Multiple hours of submissions.

Shen Qi strode away and saw Mu Rong outside the examination room.

"This... Shen Qi, are you too fast?" Mu Rong was a little surprised.

Shen Qi was a little calm: "Generally average, the third in Asia."

"Who are the top two in Asia?" Mu Rong couldn't help but laugh.

Shen Qi said solemnly, "It's the principal and Teacher Mu."

"Haha." Mu Rong finally couldn't help laughing, and felt puzzled again: "You are politically conscious, you can't get a B in the political exam, right?" She also heard that Shen Qi got a B in the political exam, and the rest Doors are all A.

"Hey, who hasn't been young and ignorant." Shen Qi was upset when he thought about the political exam, black history, a stain that cannot be erased.Of course, before the second semester of high school, Shen Qi had a lot of black history. Since it couldn't be erased, let's face it calmly.

Shen Qi stretched: "Yawn... so sleepy, I'm going to sleep."

"Go, go home and have a good rest, it's hard work." Mu Rong didn't discuss any issues with Shen Qi, and Shen Qi's actions explained everything.

The grading of the preliminary round will be completed within one to two weeks, and the CPhO rematch is scheduled to be held in mid-April.

The next day, Shen Qi met several teammates of the school sports team at Nangang No. [-] Middle School.

"I've finished the assessment, and I think I can get a 140?" said Lin Xingyu, a teammate of the Wushu team.

"I got one multiple-choice question, two fill-in-the-blank questions, and two calculation questions wrong. The score in the preliminary round was between 120 and 130." Another teammate, Bai Lingye, said.

"What about you, Mr. Shen Qi?" Lin Xingyu asked.

"I estimate it to be around 200 points? Maybe one or two points will be deducted for some steps of the last calculation question. I'm not sure, I don't know." Shen Qi said.

"Cow!" Lin Xingyu had a blind admiration for Shen Qi, and 200 points was a perfect score.

"It's over, it's over, Shen Qi has raised the qualifying score for the preliminary round." Bai Lingye said bitterly.

"Bai Lingye, I'm afraid you failed in math." Lin Xingyu said, "There are so many contestants in the country participating, and a perfect score will raise the score line? You can't complain, bad review!"

……

……

(Yesterday, on the second day of the second day of the first year, I was drinking too much at a relative's house. I just woke up and my head hurt. I got up and wrote a chapter to save the manuscript and then went back to sleep. I guess I can't write a few words today. I have to go to other relatives' houses for dinner in the evening. Men over the age of 20 are not allowed to drink alcohol.

I received a notice from the starting point, and it will be on the shelves on March 3. I have to save some manuscripts. It is estimated that it is difficult to save a chapter manuscript today, so I can only update one chapter. )
(End of this chapter)