Physics 121: Electricity & Magnetism Lecture 1

Transcription

1 Phsics 121: Electicit & Magnetism Lectue 1 Dale E. Ga Wenda Cao NJIT Phsics Depatment

2 Intoduction to Clices 1. What ea ae ou?. Feshman. Sophomoe C. Junio D. Senio E. Othe

3 Intoduction to Clices 2. How man eas have ou been at NJIT?. Less than 1 ea. 1 ea C. 2 eas D. 3 eas E. Moe than 3 eas

4 Intoduction to Clices 3. What pevious Phsics couses have ou taen?. Phsics 105/106. Phsics 111 C. Tansfe cedit fom anothe college D. Othe

5 Clices 4. What do ou thin of the idea of using clices in this class?. I lie it!. Hmm, sounds inteesting. I ll t it. C. I have to be convinced. D. Sounds boing. I doubt that I will lie it. E. I hate it.

6 ESP 5. I am thining of a lette fom to E. See if ou can pic up m thoughts.. I am thining of.. I am thining of. C. I am thining of C. D. I am thining of D. E. I am thining of E.

7 ESP2 6. Let s do it again, but show the gaph while ou select ou answe. I am thining of a lette fom to E. What lette am I thining of?. I am thining of.. I am thining of. C. I am thining of C. D. I am thining of D. E. I am thining of E.

8 Vecto asics We will be using vectos a lot in this Was of witing vecto notation couse. F ma Remembe that vectos have both F ma magnitude and diection e.g. a, θ F ma You should now how to find the components of a vecto fom its magnitude and diection a a a cosθ a asinθ a θ You should now how to find a vecto s a magnitude and diection fom its components 2 2 a a a θ tan 1 a / a

9 Poection of a Vecto and Vecto Components When we want a component of a vecto along a paticula diection, it is useful to thin of it as a poection. The poection alwas has length a cos θ, whee a is the length of the vecto and θ is the angle between the vecto and the diection along which ou want the component. You should now how to wite a vecto in unit vecto notation a a i a o a ( a, a ) acosθ sinφ θ θφ a cosθ

10 Poection of a Vecto in Thee Dimensions n vecto in thee dimensions can be poected onto the - plane. The vecto poection then maes an angle φ fom the ais. Now poect the vecto onto the ais, along the diection of the ealie poection. a θ φ The oiginal vecto a maes an angle θ fom the ais.

11 Vecto asics You should now how to genealie the case of a 2-d vecto to thee dimensions, e.g. 1 magnitude and 2 diections a, θ, φ Convesion to,, components a asinθ cosφ a a asinθ sinφ a cosθ Convesion fom,, components a a 2 a 2 a 2 a θ φ θ cos φ tan 1 1 a a / a / a Unit vecto notation: a a i a a

12 Seeing in 3 Dimensions 7. Which of these show the pope poection of the vecto onto the ais?. I.. II. C. III. D. IV. E. None of the above. θ I. II. φ III. θ IV. a a φ θ θ a a φ φ

13 Note bout Right-Hand Coodinate Sstems thee-dimensional coodinate sstem MUST obe the ight-hand ule. Cul the finges of ou RIGHT HND so the go fom to. You thumb will point in the diection.

14 Right Handed Coodinate Sstems 8. Which of these coodinate sstems obe the ight-hand ule?. I and II.. II and III. C. I, II, and III. D. I and IV. E. IV onl. I. II. III. IV.

15 Vecto Math Vecto Invese Just switch diection Vecto ddition Use head-tail method, o paallelogam method Vecto Subtaction Use invese, then add Vecto Multiplication Two inds! Scala, o dot poduct Vecto, o coss poduct Vecto ddition b Components ( ) i ( ) ( )

16 Poection of a Vecto: Dot Poduct The dot poduct sas something about how paallel two vectos ae. The dot poduct (scala poduct) of two vectos can be thought of as the poection of one onto the diection of the othe. cosθ i cosθ Components ( cosθ ) θ ( cosθ )

17 Poection of a Vecto: Dot Poduct The dot poduct sas something about how paallel two vectos ae. The dot poduct (scala poduct) of two vectos can be thought of as the poection of one onto the diection of the othe. cosθ i cosθ Components π/2 Poection is eo

18 Deivation How do we show that? Stat with Then ut So i i ) ( ) ( ) ( ) ( ) ( i i i i i i 1 1; 1; 0 0; 0; i i i i i i

19 Coss Poduct The coss poduct of two Recall angula momentum vectos sas something about L p how pependicula the ae. Toque You will find it in the contet τ F of otation, o twist. sinθ sinθ Diection pependicula to both and (ight-hand ule) θ Components (mess) i sinθ ( ) i ( ) ( )

20 Deivation How do we show th at? Stat with Then ut So i i ) ( ) ( ) ( ) ( ) ( i i i i i i 0 0; 0; ; ; i i i i i i i i i i ) ( ) ( ) (

21 Scala Fields scala field is ust one whee a quantit in space is epesented b numbes, such as this tempeatue map. Hee is anothe scala field, height of a mountain. Contous fa apat Contous close togethe steepe Contous flatte Side View

22 Gadients and Gavit Height contous h, ae popotional to potential eneg U mgh. If ou move along a contou, ou height does not change, so ou potential eneg does not change. If ou move downhill, on sa a 6% gade, it means the slope is 6/100 (fo eve 100 m of hoiontal motion, ou move downwad b 6 m). 6 Gade and gadient mean the same thing. 6% gade is a gadient of lim Δh / Δ dh / d 0.06 Δ F du / dl d mgh / dl mg dh / dl F mg sinθ dl dh θ mg

23 Vecto Fields vecto field is one whee a quantit in space is epesented b both magnitude and diection, i.e b vectos. The vecto field beas a close elationship to the contous (lines of constant potential eneg). The steepe the gadient, the lage the vectos. The vectos point along the diection of steepest descent, which is also pependicula to the lines of constant potential eneg. Imagine ain on the mountain. The vectos ae also steamlines. Wate unning down the mountain will follow these steamlines. Side View

24 nothe Eample Pessue

25 nothe Eample Pessue

26 Suface vs. Volume Vecto Fields In the eample of the mountain, note that these foce vectos ae onl coect when the obect is ON the suface. The actual foce field anwhee othe than the suface is evewhee downwad (towad the cente of the Eath. The suface ceates a nomal foce evewhee nomal (pependicula) to the suface. The vecto sum of these two foces is what we ae showing on the contou plot. Side View

27 Vecto Field Due to Gavit When ou conside the foce of Eath s gavit in space, it points evewhee in the diection of the cente of the Eath. ut emembe that the stength is: GMm F 2 This is an eample of an invese-squae foce (popotional to the invese squae of the distance).

28 Idea of Test Mass Notice that the actual amount of foce depends on the mass, m: GMm F 2 It is convenient to as what is the foce pe unit mass. The idea is to imagine putting a unit test mass nea the Eath, and obseve the effect on it: F GM g( ) 2 m g() is the gavitational field.

29 Meaning of g() F GM 9. What ae the units of g( ) 2?. Newtons/mete (N/m). Metes pe second squaed (m/s 2 ) C. Newtons/ilogam (N/g) D. oth and C E. Fulongs/fotnight m

30 Meaning of g() 10. What is anothe name fo g() in F GM g( ) 2? m. Gavitational constant. Gavitational eneg C. cceleation of gavit D. Gavitational potential E. Foce of gavit

31 Gavitational Field We can theefoe thin of the action-at-a-distance of gavit as a field that pemeates all of space. We daw field lines that show both the diection and stength of the field (fom the densit of field lines). The field cannot be seen o touched, and has no effect until ou conside a second mass. What happens if we have two equal masses? Supeposition ust vecto sum the two fields.

32 Gavitational Field of Two Equal Masses gain, thin of adding a small test mass. The foce vectos show the diection and stength of the foce on such a test mass. We can daw field lines that follow the foce vectos. We will be using these same concepts when we tal about electic chage in Lectue 2, and the electic field in Lectue 3.