《物理双语教学课件》Chapter 16 Gauss' Law 高斯定理

Chapter 16 Gauss’ Law

If you want to find the center of mass of a potato, you can do so by experiment or by laborious calculation, involving the numerical evaluation of a triple integral. However, if the potato happens to be a uniform ellipsoid, you know from its symmetry exactly where the center of mass is without calculations. Such are the advantages of symmetry. Symmetrical situations arise in all areas of physics; when possible, it makes sense to cast the laws of physics in forms that take full advantage of this fact.

Gauss’s law can be used to take advantage of special symmetry situation. For electrostatics problems, it is the full equivalent of Coulomb’s law.

Central to Gauss’ law is a hypothetical closed surface called a Gaussian surface. The Gaussian surface can be of any shape you wish to make it, but the most useful surface is one that mimics the symmetry of the problem at hand. Thus the Gaussian surface will often be a sphere, a cylinder, or some other symmetrical form. It must be closed surface, so that a clear distinction can be made between points that are inside the surface, and outside the surface. Gauss’ law relates the electric fields at points on a Gaussian surface and the net charge enclosed by that

surface .

16.1 Flux of an Electric Field

1. To determine the flux of an

electric field , consider figure (a),

which shows an arbitrary

Gaussian surface immersed in a

non-uniform electric field. Let us

divide the surface into small

squares of area A ?, each square

being small enough to permit us to neglect any curvature and consider the individual square to be flat. We represent each

such element of area with an area vector A ?, whose

magnitude is the area A ?. Each vector A ? is perpendicular to

the Gaussian surface and directed away from the interior of the surface.

2. Because the squares have been taken to be arbitrarily small, the electric field E may be taken as constant over any given

square. The vectors A ? and E for each square then make

some angle θ with each other. A provisional definition for the flux of the electric field for the Gaussian surface of above figure is A E ??∑=Φ. This equation instructs us to visit each square on the Gaussian surface, to evaluate the scalar product

A E ??∑ for the two vectors E and A ? that we find there, and to sum the results algebraically for all the squares that make up the surface.

3. The exact definition of the flux of the electric field through a closed surface is found by allowing the area of the squares shown in above figure (a) to become smaller and smaller, approaching a differential limit dA. The area vectors then approach a differential limit A d . The sum of the above equation then becomes an integral and we have, for the definition of electric flux, ??=ΦA d E . The circle on the integral sign indicates that the integration is to be taken over the entire (closed) surface. The flux of the electric field is a scalar, and its SI unit is the newton-square-meter per coulomb ()/2C m N ?.

4. We can interpret above definition in the following way: First recall that we can use the density of electric field lines passing through an area as a measure of an electric field E there. Specifically, the magnitude E is proportional to the number of electric field lines per unit area. Thus, the dot product A d E ? is proportional to the number of electric field lines passing through area A d . Then, because the integration is carried out over a Gaussian surface, which is closed, we see that the electric flux

Φ through a Gaussian surface is proportional to

the net number of electric field lines passing through that surface.

16.2 Gauss ’ Law

1. Gauss ’ law relates the net flux Φ of an electric field through a closed surface to the net charge enc q that is enclosed by that

surface . It tells us that )'(00law Gauss q A d E enc ?=?=Φ εε. Above equation hold only when the net charge is located in a vacuum or in air . The net charge enc q is the algebraic sum of all the

enclosed positive and negative charges, and it can be positive, negative, or zero. If

enc q is positive, the net flux is outward; if enc q is negative, the flux is inward.

2. Charge outside the surface, no matter how large or how close it may be, is not included in the term enc q in Gauss ’ law. The

exact form or location of the charges inside the Gauss surface is also of no concern; the

only things that matter are

the magnitude and sign of

the net enclosed charge.

3. The E on the left side,

however, is the electric field

resulting from all charges ,

both those inside and those outside the Gaussian surface.

4.See right figure.

16.3 A Charged Isolated Conductor

1.Gauss’law permits us to prove an important theorem about

isolated conductors: if an excess charge is placed on an isolated conductor, that amount of charge will move entirely to the surface of the conduct. None of the excess charge will be found within the body of the conductor. This might seem reasonable, considering that charge with the same sign repel each other. You might imagine that, by moving to the surface, the added charge are getting as far away from each other as they can. See Figure (a).

2.An isolated conductor

with a cavity: Figure (b)

shows the same

conductor, but now with

a cavity that is totally within the conductor. The distribution of

charge or the pattern of the electric field will not be charged anymore.

3.The external electric field: T he electric field E at and just

outside the conductor’s surface must be perpendicular to that

surface, and its magnitude is 0εσ

=E , in which σ is the charge

per unit area. It means t he magnitude of the electric field at a location just outside a conductor is proportional to the surface charge density at that location on the conductor . If the charge on the conductor is positive, the electric field points away from the conductor; it points toward the conductor if the charge is negative.

16.4 Applying Gauss ’ Law

1. Cylindrical symmetry

(1) Figure shows a section of an infinitely

long cylindrical plastic rod with a uniform

(positive) lines charge density λ. The

expression for the magnitude of the electric field

E at a distance r from the axis of the rod is

r E 02πελ=. (2) The direction of E is radially outward if the charge is positive; and radially inward if it is negative.

2. Planer symmetry

(1) Figure shows a portion of a thin,

infinite, non-conducting sheet with a

uniform (positive) surface charge σ.

The magnitude of the electric field is 02εσ

=E , and its

direction is perpendicular to and away from the sheet. Since we are considering an infinite sheet with uniform charge density, this result holds for any point at a finite distance from the sheet.

(2) One conducting plate:

(3) Two conducting plates:

3. Spherical symmetry : (1) We can use Gauss ’ law to prove the two shell theorems: (a) A shell of uniform charge attracts or repels a charged particle that is outside the shell as if all the shell ’s charge were concentrated at the center of the shell . (b)

A shell of uniform charge exerts no electrostatic force on a charged particle that is located inside the shell .

chapter2 习题答案

一．名词解释 1．假根 是Rhizopus(根霉属)等低等真菌匍匐菌丝与固体基质接触处分化出来的根状结构，具有固着和吸取养料等功能。 2．假菌丝 当酵母菌进行一连串的芽殖后，如果长大的子细胞与母细胞不立即分离，其间仅以狭小的面积相连，则这种藕节状的细胞串就称为假菌丝 3．气生菌丝 伸展到空间的菌丝体，颜色较深、直径较粗的分枝菌丝，其成熟后分化成孢子丝4．子囊果 能产生有性孢子的、结构复杂的子实体称为子囊果 5．生活史 又称生命周期，指上一代生物经一系列生长、发育阶段而产生下一代个体的全部过程为生活史 6、异宗配合：指毛霉在形成接合孢子时，凡是由不同性的菌丝体上形成的性器官结合 而产生有性孢子的则称异宗配合。 7、同宗配合：指毛霉在形成接合孢子时，凡是由同一个菌丝体上形成的配子囊结合而 产生有性孢子的则称同宗配合。 8、锁状联合：指担子菌亚门的次生菌丝的菌丝尖端生长方式。 9Saccharomyces cerevisiae 酿酒酵母 二．填空 1、真菌的无性繁殖方式有裂殖、芽殖、无性孢子繁殖和菌丝体断裂。 2、酵母菌的无性繁殖方式主要有裂殖、芽殖。 3、真菌的有性孢子的种类有：卵孢子、接合孢子、子囊孢子和担孢子四种；真菌的无 性孢子的种类有：游动孢子、子囊孢子、分生孢子、节孢子和厚垣孢子五种。 4、根霉的形态特征是具有假根和匍匐功丝且菌丝无隔；曲霉的形态特征是具顶囊和足 细胞，菌丝有隔；青霉的形态特征是具扫帚状的分生孢子梗。 5．粘菌可分为__细胞粘菌___和_原质团粘菌__两个门. 6．真核微生物细胞质核糖体类型为80S，原核微生物核糖体类型为_70S___. 7．子囊果有闭囊果，_子囊壳_____ 和_____子囊盘________三种类型。

小王子The Little Prince 英文Chapter 16

Chapter 16 ? the narrator discusses the Earth's lamplighters So then the seventh planet was the Earth. The Earth is not just an ordinary planet! One can count, there 111 kings (not forgetting, to be sure, the Negro kings among them), 7000 geographers, 900,000 businessmen, 7,500,000 tipplers, 311,000,000 conceited men?? that is to say, about 2,000,000,000 grown?ups. To give you an idea of the size of the Earth, I will tell you that before the invention of electricity it was necessary to maintain, over the whole of the six continents, a veritable army of 462,511 lamplighters for the street lamps. Seen from a slight distance, that would make a splendid spectacle. The movements of this army would be regulated like those of the ballet in the opera. First would come the turn of the lamplighters of New Zealand and Australia. Having set their lamps alight, these would go off to sleep. Next, the lamplighters of China and Siberia would enter for their steps in the dance, and then they too would be waved back into the wings. After that would come the turn of the lamplighters of Russia and the Indies; then those of Africa and Europe, then those of South America; then those of South America; then those of North America. And never would they make a mistake in the order of their entry upon the stage. It would be magnificent. Only the man who was in charge of the single lamp at the North Pole, and his colleague who was responsible for the single lamp at the South Pole?? only these two would live free from toil and care: they would be busy twice a year.

HALCON算子函数Chapter 16：System

HALCON算子函數——Chapter 16 : System 16.1 Database 1. count_relation 功能：在HALCON數據庫中實體的數目。 2. get_modules 功能：查詢已使用模塊和模塊關鍵碼。 3. reset_obj_db 功能：HALCON系統的初始化。 16.2 Error-Handling 1. get_check 功能：HALCON控制模式的說明。 2. get_error_text 功能：查詢HALCON錯誤測試後錯誤數目。3. get_spy 功能：HALCON調試工具當前配置。 4. query_spy 功能：查詢HALCON調試工具可能的設置。5. set_check 功能：激活和鈍化HALCON控制模式。

6. set_spy 功能：HALCON調試工具的控制。 16.3 Information 1. get_chapter_info 功能：獲取程序有關章節的信息。 2. get_keywords 功能：獲取指定給程序的關鍵字。 3. get_operator_info 功能：獲取關於HALCON程序的信息。 4. get_operator_name 功能：獲取由給定字符串作為它們的名字的程序。 5. get_param_info 功能：獲取關於程序參數的信息。 6. get_param_names 功能：獲取一個HALCON程序參數的名字。 7. get_param_num 功能：獲取一個HALCON程序不同參數類的數目。 8. get_param_types 功能：獲取一個HALCON程序控制參數的缺省數據類型。

抽象代数Chapter16习题答案

Math 5286H:Fundamental Structures of Algebra II HW 5Solutions,(April 27th,2012) Problems from Chapter 16of Artin’s Algebra: 9.3As discussed in class and in the book (Examples 16.9.2(c)),if α= 7,then letting K be the splitting ?eld of α’s irreducible polynomial,we have that Gal (K/Q )～=V 4=C 2×C 2.Consequently,K has degree 4over Q ,and K =Q (α).K also contains three di?erent quadratic extensions (over Q )as sub?elds.Since a quadratic extension over Q looks like Q (√a )and Q (√a +√a +√a ′+√4? √α2±2αα′+α′2=α±α′∈K ,thus 7)±2√7)=√2and √2+√28,we see this equals 4α,thus α= 2+ 2. In fact √14 2+α?α′ D ∈Q .One could also compute the roots of x 4+1explicitly and compute D = 1≤i

Chapter16-18英语词汇记忆法

chapter 16 adept adept熟练的(ad+ept能力=有做事能力=熟练的) ad-加在在单词或词根前,表示”做…,加强…” adapt适应(ad+apt能力=有适应能力) adept熟练的(ad+ept能力=有做事能力=熟练的) adopt收养,采纳(ad+opt选择=选出来=采纳) adhere坚持(ad+here粘=粘在一起=坚持) adjacent邻近的(ad+jacent躺=躺在一起=邻近的) adjoin贴近,毗连(ad+join参加=参加在一起=贴近) administrate管理,执行(ad+ministr部长+ate=做部长=管理) admire羡慕(ad+mire惊奇=惊喜,羡慕) adumbrate预示(ad+umbr影子

+ate=[将来的]影子出现=预示) adjust调整(ad+just+正确=弄正确=调整) adventure冒险(ad+venture冒险) admonish告诫,警告(ad+mon警告 +ish=一再警告) advent来临,来到(ad+vent来=来到) adept熟练的,老练的(ad+ept=有能力=熟练的) apt,ept=fit, ability,表示”适应,能力”apt有倾向的,适当的 aptitude适应能力,才能(apt+itude表示状态=有能力的状态) aptly适当地(apt+ly) adapt使适应,适应(环境)(ad一再+apt=一再适应(环境)) adaptable有适应能力的(adapt适应 +able能…的) adaptation适应,(书)改写本

(adapt+ation=适应=适应(各种人)的书=改写本) inapt不适当的,无能的(in无+apt能力) adept熟练的,老练的(ad+ept=有能力=熟练的) encompass包围,围绕(en+compass 包围) em-,en- ①表示”进入…之中,包围” embrace拥抱(em+brace胳膊=进入怀抱) embed安置,嵌于(em+bed范围,床=进入范围=安置) empathy感情相融的(em+pathy感情=进入感情) embarrass使难堪(em+barrass套子=进入套子=难堪) embattle整军备战(em+battle战斗) enroll注册,记入名册(en+roll名单) encage关入笼中(en+cage笼子)

7下英语读本 Chapter 16 - 教师版

Chapter 16 诵读 What a wonderful world 多么美好的世界 I see trees of green, red roses, too I see them bloom for me and you And I think to myself What a wonderful world 我看到绿色的树，红色的玫瑰，太 我看到它们为我和你绽放 我对自己说 多么美好的世界 I see skies of blue and clouds of white The bright blessed day The dark sacred night And I think to myself What a wonderful world 我看到蓝天和白云 幸福的日子 神圣的黑夜 我对自己说 多么美好的世界 The colours of the rainbow So pretty in the sky Are also on the faces of people going by I see friends shaking hands Saying “How do you do?” They’re really saying “I love you.” 彩虹的颜色 天空如此美丽 路过的人脸上也有吗 我看到朋友们在握手 说“你好吗?” 他们真的在说“我爱你”。 I hear babies cry I watch them grow They’ll learn much more Than I’ll never know And I think to myself What a wonderful world 我听到婴儿哭泣

宏经chapter 16

SOLUTIONS TO TEXT PROBLEMS: chapter 16 Quick Quizzes 1. The three functions of money are: (1) medium of exchange; (2) unit of account; and (3) store of value. Money is used as a medium of exchange because money is the item people use to purchase goods and services. Money is used as a unit of account because it is the yardstick people use to post prices and record debts. Money is used as a store of value because it is an item people use to transfer purchasing power from the present to the future. 2. The primary responsibilities of the Federal Reserve are to regulate banks, ensuring the health of the banking system, and to control the quantity of money that is made available in the economy. If the Fed wants to increase the supply of money, it usually does so by creating dollars and using them to purchase government bonds from the public in the nation’s bond markets. 3. Banks create money when they make loans and hold a fraction of the amount of the loans in reserves, resulting in an expansion of both money and credit in the economy. If the Fed wanted to use all three of its tools to decrease the money supply, it would: (1) sell government bonds from its portfolio in the open market to reduce the number of dollars in circulation; (2) increase reserve requirements to reduce the money created by banks; and (3) increase the discount rate to discourage banks from borrowing reserves from the Fed. Questions for Review 1. Money is different from other assets in the economy because it is the most liquid asset available. Other assets vary widely in their liquidity. 2. Commodity money is money with intrinsic value, like gold, which can be used for purposes other than as a medium of exchange. Fiat money is money without intrinsic value; it has no value other than its use as a medium of exchange. Our economy today uses fiat money. 3. Demand deposits are balances in bank accounts that depositors can access on demand simply by writing a check. They should be included in the stock of money because they can be as a medium of exchange. 4. The Federal Open Market Committee (FOMC) is responsible for setting monetary policy in the United States. The FOMC consists of the seven members of the Federal Reserve Board of Governors and five of the 12 presidents of Federal Reserve Banks. Members of the Board of Governors are appointed by the president of the United States and confirmed by the U.S. Senate. The presidents of the Federal Reserve Banks are chosen by each bank’s board of directors. 5. If the Fed wants to increase the supply of money with open-market operations, it purchases U.S. government bonds from the public on the open market. The purchase increases the number of dollars in the hands of the public, thus raising the money supply. 6. Banks do not hold 100 percent reserves because it is more profitable to use the reserves to make loans, which earn interest, instead of leaving the money as reserves, which earn no interest. The amount of reserves banks hold is related to the amount of money the banking system creates through the money multiplier. The smaller the fraction of reserves banks hold, the larger the money multiplier, since each dollar of reserves is used to create more money. 7. The discount rate is the interest rate on loans that the Federal Reserve makes to banks. If the Fed raises the discount rate, fewer banks will borrow from the Fed, so banks' reserves will be lower,

香港朗文3Achapter16单词与句子

3A Chapter One Weather and seasons 1. Words and phrases: (单词和短语) ①weather 天气 foggy有雾的windy刮风的rainy下雨的cloudy多云的sunny 晴朗的 snowy下雪的 dry干燥的wet潮湿的 warm温暖的hot热的cool凉的cold冷的 ②season季节 spring春天summer夏天autumn秋天winter冬天favourite最喜欢的 ③a bit 有点儿chat 聊天strong 强壮的feel 感觉warmly 温暖地 outside在外面 ④plant flowers种花go to the park去公园eat ice cream吃冰淇琳 go swimming去游泳fly kites放风筝go hiking去远足 wear a scarf带围巾eat hotpot吃火锅 put on穿上take off脱掉give up 放弃 button up 扣上blow off吹掉blow hard大力吹 play in the rain在雨中玩wear a hat戴帽子 2. sentences: (句子)

注：①weather 天气、②season季节部分的单词和句子都要求学生能听说读写。其他③、 ④部分的单词和词组要求学生能认读。 3A Chapter Two Festivals we like 1. Words and phrases: (单词和短语) ①festival 节日Christmas 圣诞节Chinese New Year 春节

Mid-Autumn Festival 中秋节Easter 复活节Dragon Boat Festival 端午节 ②send card 送卡片give presents 送礼物watch fireworks 看烟花 get lucky money得压岁钱eat moon cakes 吃月饼eat chocolate eggs吃巧克力蛋 watch dragon boat races 看龙舟赛 ③Santa Claus 圣诞老人already 已经reindeer驯鹿find 找到need 需要person 人sick 生病try to help 试着帮忙start 开始sink下沉worry担心idea主意pull拉enjoy 享受last year 去年 ④poem 诗fun乐趣fresh 新鲜bright 明亮cheer欢呼bloom 开花humid 潮湿cookies曲奇饼soggy 潮乎乎drum 鼓shine照耀 inside里面lanterns 灯笼wait for等待full moon圆月 hope 希望soon 不久excited兴奋end结束 2. Sentences: (句子) ① Which festival do you like best? 你最喜欢哪个节日？ I like Mid-Autumn Festival best. 我最喜欢中秋节。 We eat moon cakes at Mid-Autumn Festival. 我们在中秋节吃月饼。 ② In spring, it is warm and foggy. 春天天气温暖多雾。 Chinese New Year is in spring. 春节在春天。 People watch fireworks at Chinese New Year. 人们在春节看烟花。 ③ Don’t worry. 别担心。 Never mind. 没有关系。 Thank you for your help. 谢谢你的帮助。 They’re racing. 他们在比赛。 We need one more person. 我们还需要一个人。 One of our friends is sick. 我们的一个朋友生病了。 注: Words and phrases (单词和短语)①②的单词和句子①②要求学生能听、说、读、写，必须掌握。