Introduction: Vectors and Integrals

Transcription

1 Intoduction: Vectos and Integals

2 Vectos a Vectos ae chaacteized by two paametes: length (magnitude) diection a These vectos ae the same Sum of the vectos: a b a a b b a b a b a

3 Vectos Sum of the vectos: fo a lage numbe of vectos the pocedue is staightfowad a a b a a b c c b c a Vecto ca (whee c is the positive numbe) has the same diection as a, but a a a a a c its length is times lage Vecto (whee is the negative numbe) has the diection opposite to, c ca c a and times lage length a a

4 Vectos The vectos can be also chaacteized by a set of numbes (components), i.e. a ( a, a,...) This means the following: if we intoduce some basic vectos, fo example x and y in the plane, then we can wite a ax a y x, y usually have unit magnitude y x a a y ax Then the sum of the vectos is the sum of thei components: a b a ( a, a) ( b, b ) ( a, a ) ab ( a b, a b ) ca ( ca, ca )

5 Vectos: Scala and Vecto Poduct a b Scala Poduct a b is the scala (not vecto) abcos( ) If the vectos ae othogonal then the scala poduct is 0 a b a b 0 c a b Vecto Poduct a b c is the VECTOR, the magnitude of which is c Vecto is othogonal to the plane fomed by and b absin( ) If the vectos have the same diection then vecto poduct is 0 a b ab 0 a

6 Vectos: Scala Poduct a b Scala Poduct a b is the scala (not vecto) abcos( ) If the vectos ae othogonal then the scala poduct is 0 a b a b 0 y x ax a a x a y a a y fom the definition of the scala poduct It is staightfowad to elate the scala poduct of two vectos to thei components in othogonal basis x, y If the basis vectos ae othogonal and have unit magnitude (length) then we can take the scala poduct of vecto and basis vectos : a ax ay x, y acos( ) ax a xxa y x a = (unit magnitude) =0 (othogonal) acos( / ) asin( ) a y ax y a y y a

7 a b a b a b a a a a y a x ax a y a a a ( a, a) a x a y a b ab abcos( ) a b a b 0 c a b c a b c absin( ) a b a b 0

8 Vectos: Examples b a a c b a The magnitude of a is 5 What is the diection and the magnitude of b 0.a The magnitude of b is b 0.5, the diection is opposite to a The magnitude of a is 5, the magnitude of b is, the angle is /3 What is the scala and vecto poduct of a and b a b 5cos( /3) 5 b a b c c 5sin( /3) 5 3

9 Integals Basic integals: b b dx n n n n x n a b n n xdx b a a n a You need to ecognize these types of integals. Examples: b dx ( x c) a n intoduce new vaiable y x c dy dx b a bc dx dy n n ( x c) y ac Impotant: Diffeent Limits in the Integals b x a xdx ( c) n intoduce new vaiable y x c dy xdx b b c xdx ( ) n x c y a a c dy n

10 Integals Integals containing vecto functions b E () tdt a a E() t E() t t y x b How can we find the values of such integals? b E () tdt a - this is the vecto, so we can calculate each component of this vecto E() t E () t x E () t y x, y We can wite, whee only scala functions depend on t, but not the basis vectos then integal takes the fom Then the integal takes the fom E() tdt x E() tdt y E() tdt b b b a a a E(), t E() t so now thee ae two integals which contain only scala functions

11 0 Integals Example: ( ) ( )cos( ) d ( ) - along the adius, then we can wite the adial vecto in tems of adius ( ) ( ) x ( ) y cos( ) x sin( ) y y x Then we have the following expession fo the integal ( )cos( ) d x cos ( ) d y cos( )sin( ) d x [ cos( )] d 0 0 sin( ) d 0

12 Chapte 5 Electicity and Magnetism Electic Fields: Coulomb s Law Reading: Chapte 5

13 Electic Chages Thee ae two kinds of electic chages - Called positive and negative Negative chages ae the type possessed by electons Positive chages ae the type possessed by potons Chages of the same sign epel one anothe and chages with opposite signs attact one anothe Electic chage is always conseved in isolated system Neutal equal numbe of positive and negative chages Positively chaged 3

14 Electic Chages: Conductos and Isolatos Electical conductos ae mateials in which some of the electons ae fee electons These electons can move elatively feely though the mateial Examples of good conductos include coppe, aluminum and silve Electical insulatos ae mateials in which all of the electons ae bound to atoms These electons can not move elatively feely though the mateial Examples of good insulatos include glass, ubbe and wood Semiconductos ae somewhee between insulatos and conductos 4

15 Electic Chages

16 Electic Chages

17 Electic Chages

18 Electic Chages

19 Electic Chages

20 Electic Chages

21 Electic Chages

22 Electic Chages

23 Electic Chages

24 Electic Chages

25 Electic Chages

26 Electic Chages

27 Electic Chages Thee ae two kinds of electic chages - Called positive and negative Negative chages ae the type possessed by electons Positive chages ae the type possessed by potons Chages of the same sign epel one anothe and chages with opposite signs attact one anothe Electic chage is always conseved in isolated system Neutal equal numbe of positive and negative chages Positively chaged 7

28 Electic Chages: Conductos and Isolatos Electical conductos ae mateials in which some of the electons ae fee electons These electons can move elatively feely though the mateial Examples of good conductos include coppe, aluminum and silve Electical insulatos ae mateials in which all of the electons ae bound to atoms These electons can not move elatively feely though the mateial Examples of good insulatos include glass, ubbe and wood Semiconductos ae somewhee between insulatos and conductos 8

29 Consevation of Chage Electic chage is always conseved in isolated system Two identical sphee q μc q μc They ae connected by conducting wie. What is the electic chage of each sphee? The same chage q. Then the consevation of chage means that : q q q q q q μc 0.5μC Fo thee sphees: q μc q μc q3 3 μc 3q qq q3 qq q3 3 q μc μc 3 3 9

30 Coulomb s Law Mathematically, the foce between two electic chages: qq F ˆ ke The SI unit of chage is the coulomb (C) k e is called the Coulomb constant k e = x 0 9 N. m /C = /(4πe o ) e o is the pemittivity of fee space e o = x 0 - C / N. m Electic chage: electon e = -.6 x 0-9 C poton e =.6 x 0-9 C 30

31 Coulomb s Law F F k q q e Diection depends on the sign of the poduct F F opposite diections, the same magnitude F qq qq 0 F F qq 0 F F qq 0 F The foce is attactive if the chages ae of opposite sign The foce is epulsive if the chages ae of like sign q q Magnitude: F F k e 3

32 Coulomb s Law: Supeposition Pinciple The foce exeted by q on q 3 is F 3 The foce exeted by q on q 3 is F 3 The esultant foce exeted on q 3 is the vecto sum of F 3 and F 3 3

33 q q Coulomb s Law qq F k ˆ e μc q μc 3 3 μc 5m 6m F F Magnitude: Resultant foce: F F F 3 3 F F 3 3 F q3 q F3 ke N 5 0 N q3 q F3 ke N.0 N F F F N N (50.0 ).80 33

34 q q Coulomb s Law qq F k ˆ e μc q μc 3 3 μc 5m F F 6m 3 3 Magnitude: Resultant foce: F F F3 F F 6 6 q3 q F3 ke N 5 0 N q q F ke N 7.0 N F F F N N 3 ( ) F 3 34

35 q q Coulomb s Law qq F k ˆ e μc q μc 3 3 μc Magnitude: F F 5m 6m 3 3 Resultant foce: F FF3 F F 6 6 q3 q F3 ke N.0 N 6 6 q q F ke N 7.0 N F F F N N 3 ( ) 5 0 F 3 35

36 Coulomb s Law qq F k ˆ q q e μc q μc 3 3 F μc Magnitude: q q F ke N 5 0 N 6 5m 6 6 q3 q F3 ke N.0 N 5 3 5m F 3 7m 6m F 3 F F 3 3 7m 3 F 5m F 3 6m 7m F Resultant foce: F FF3 F 3 F 6m 3 F F F 3 F 36 F

37 Chapte 5 Electic Field 37

38 Electic Field An electic field is said to exist in the egion of space aound a chaged object This chaged object is the souce chage When anothe chaged object, the test chage, entes this electic field, an electic foce acts on it. The electic field is defined as the electic foce on the test chage pe unit chage E F q 0 If you know the electic field you can find the foce F qe If q is positive, F and E ae in the same diection If q is negative, F and E ae in opposite diections 38

39 Electic Field The diection of E is that of the foce on a positive test chage The SI units of E ae N/C E F q 0 Coulomb s Law: Then qq 0 F k ˆ e F q E k ˆ e q 0 39

40 Electic Field q is positive, F is diected away fom q The diection of E is also away fom the positive souce chage q is negative, F is diected towad q E is also towad the negative souce chage qq 0 F k ˆ e F q E k ˆ e q 0 40

41 Electic Field: Supeposition Pinciple At any point P, the total electic field due to a goup of souce chages equals the vecto sum of electic fields of all the chages q E k ˆ e i i i i 4

42 Electic Field q μc q μc q E k ˆ e 5m 6m E E Electic Field: E E E E E E Magnitude: 6 q 9 0 E ke N/ C 5 0 N/ C 6 6 q 90 E ke N/ C N/ C E E E N C N C ( ) / / 4

43 Electic Field q μc q μc q E k ˆ e E 5m E 6m 7m Electic field E E E E E E Magnitude: 6 q 9 0 E ke N/ C 5 0 N/ C 6 6 q 90 E ke N/ C N/ C 5 43

44 Electic Field q E k ˆ e q 0 μc q 0μC z E Diection of electic field? 5m 5m 6m 44

45 Electic Field q E k ˆ e q 0 μc q 0μC E E Electic field E E E E E 5m 5m Magnitude: 6m 6 q E E ke N/ C N/ C 5 E E cosφ cosφ E E

46 Example q 0 μc q 0μC m m m l m l m T F E Electic field T mgf E 0 qq F ˆ E ke mg g T cos mg Tsin F E tan F E e mg k qq mg tan F k qq E e mg mg 46

47 Coulomb s Law: qq F ˆ ke 3 5m F 7m F 3 6m Resultant foce: F FF3 F 3 F F q E k ˆ e q E k ˆ e i i i i F qe E E E E E E E 5m E 6m 7m 47