PHYS 1444 Section 501 Lecture #7

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1 PHYS 1444 Section 51 Lectue #7 Wednesday, Feb. 8, 26 Equi-potential Lines and Sufaces Electic Potential Due to Electic Dipole E detemined fom V Electostatic Potential Enegy of a System of Chages Capacitos and Capacitance Today s homewok is #4, due 7pm, Thusday, Feb. 16!! 1

2 Announcements Distibution list I still have 5 of you incommucado A Lage Cloud Chambe wokshop Date and time: 1am, this Satuday, Feb. 11 Will build and opeate a pototype chambe with all the new gadgets Quiz next Monday, Feb. 13 Coves CH21 CH 23 1 st tem exam Wednesday, Feb. 22 Coves CH21 CH25 Reading assignments CH23 9 2

3 A Cloud Chambe Image of A Cosmic Tack 3

4 Equi-potential Sufaces Electic potential can be visualized using equipotential lines in 2-D o equipotential sufaces in 3-D Any two points on equipotential sufaces (lines) ae on the same potential What does this mean in tems of the potential diffeence? The potential diffeence between the two points on an equipotential suface is. How about the potential enegy diffeence? Also. What does this mean in tems of the wok to move a chage along the suface between these two points? No wok is necessay to move a chage between these two points. 4

Equi-potential Sufaces An equipotential suface (line) must be pependicula to the electic field.

5 Equi-potential Sufaces An equipotential suface (line) must be pependicula to the electic field. Why? If thee ae any paallel components to the electic field, it would equie wok to move a chage along the suface. Since the equipotential suface (line) is pependicula to the electic field, we can daw these sufaces o lines easily. Thee can be no electic field inside a conducto in static case, thus the entie volume of a conducto must be at the same potential. So the electic field must be pependicula to the conducto suface. Paallel Plate Point chages Just like a topological map 5

6 V Electic Potential due to Electic Dipoles What is an electic dipole? Two equal point chage Q of opposite sign sepaated by a distance l and behaves like one entity: PQl The electic potential due to a dipole at a point p We take V at infinity The simple sum of the potential at P by the two chages is V Qi 1 ia 1 ( Q) Q Q + + Since lcosθ and if >>l, >>, thus ~+ and Q l cosθ 1 p cosθ V by a dipole at a distance fom Wednesday, Feb. 8, 26 PHYS , the Sping dipole Q + V 1 + p cosθ 6

7 E Detemined fom V Potential diffeence between two points unde the electic field is Vb Va b E dl a So in a diffeential fom, we can wite dv E dl El dl What ae dv and E l? dv is the infinitesimal potential diffeence between two points sepaated by the distance dl E l is the field component along the diection of dl. Thus we can wite the field component E l as dv El Physical dl Meaning? The component of the electic field in any diection is equal to the negative ate of change of the electic potential as a function of distance in that diection.!! 7

8 E Detemined fom V The quantity dv/dl is called the gadient of V in a paticula diection If no diection is specified, the tem gadient efes to the diection V changes most apidly, and this would be the diection of the field vecto E at that point. dv So if E and dl ae paallel to each othe, E If E is witten as a function of x, y and z, l efes to V V V x, y and z Ex Ey E x z y z V x is the patial deivative of V with espect to x, with y and z held constant E gadv V i + j + k V In vecto fom, x y z Wednesday, Feb. 8, 26 PHYS , Sping 26 i + j + k is called the del o the x y z D. gadient Jaehoon opeato Yu and is a vecto opeato. dl 8

9 Electostatic Potential Enegy Conside a point chage q is moved between points a and b whee the electostatic potentials due to othe chages ae V a and V b The change in electostatic potential enegy of q in the field by othe chages is U U b U a qvb Va qvba Now what is the electostatic potential enegy of a system of chages? ( ) Let s choose V at infinity If thee ae no othe chages aound, single point chage Q 1 in isolation has no potential enegy and is exeted on with no electic foce 9

10 Electostatic Potential Enegy; Two chages If a second point chage Q 2 is bought close to Q 1 at the distance 12, the potential due to Q 1 at the position of Q 2 is Q V 1 12 The potential enegy of the two chages elative to V at infinity is 1 QQ U Q V This is the wok that needs to be done by an extenal foce to bing Q 2 fom infinity to a distance 12 fom Q 1. It is also a negative of the wok needed to sepaate them to infinity

11 Electostatic Potential Enegy; Thee Chages So what do we do fo thee chages? Wok is needed to bing all thee chages togethe Wok needed to bing Q 1 to a cetain place without the pesence of any chage is. 1 QQ 1 2 Wok needed to bing Q 2 to a distance to Q 1 is U Wok need to bing Q 3 to a distance to Q 1 and Q 2 is U 3 U 13 + U 23 1 QQ So the total electostatic potential of the thee chage system is QQ 1 2 QQ 1 3 QQ 2 3 U U12 U13 U V at What about a fou chage system? + 1 QQ [ ] 11

12 Electostatic Potential Enegy: electon Volt What is the unit of electostatic potential enegy? Joules Joules is a vey lage unit in dealing with electons, atoms o molecules in atomic scale poblems Fo convenience a new unit, electon volt (ev), is defined 1 ev is defined as the enegy acquied by a paticle caying the chage equal to that of an electon (qe) when it moves acoss a potential diffeence of 1V How many Joules is 1 ev then? 1eV C 1V ev howeve is not a standad SI unit. You must convet the enegy to Joules fo computations. What is the speed of an electon with kinetic enegy 5eV? J 12

13 Capacitos (o Condenses) What is a capacito? A device that can stoe electic chage But does not let them flow though What does it consist of? Usually consists of two conducting objects (plates o sheets) placed nea each othe without touching Why can t they touch each othe? The chage will neutalize Can you give some examples? Camea flash, UPS, Suge potectos, binay cicuits, etc How is a capacito diffeent than a battey? Battey povides potential diffeence by stoing enegy (usually chemical enegy) while the capacito stoes chages but vey little enegy. 13

A cylindical capacitos ae essentially paallel plates wapped aound as a

14 Capacitos A simple capacito consists of a pai of paallel plates of aea A sepaated by a distance d. A cylindical capacitos ae essentially paallel plates wapped aound as a cylinde. How would you daw symbols fo a capacito and a battey? Capacito - - Battey (+) - i- (-) Cicuit Diagam 14

15 Capacitos What do you think will happen if a battey is connected ( o the voltage is applied) to a capacito? The capacito gets chaged quickly, one plate positive and the othe negative in equal amount. Each battey teminal, the wies and the plates ae conductos. What does this mean? All conductos ae at the same potential. And? So the full battey voltage is applied acoss the capacito plates. So fo a given capacito, the amount of chage stoed in the capacito is popotional to the potential diffeence V ba between the plates. How would you wite this fomula? Q CV ba C is a popety of a capacito so does not depend on Q o V. C is a popotionality constant, called capacitance of the device. 15 What is the unit? C/V o Faad (F) Nomally use µf o pf.

Detemination of Capacitance C can be detemined analytically fo capacitos w/ simple geomety and ai in between. Let s conside a paallel plate capacito. Plates have aea A each and sepaated by d.

16 Detemination of Capacitance C can be detemined analytically fo capacitos w/ simple geomety and ai in between. Let s conside a paallel plate capacito. Plates have aea A each and sepaated by d. d is smalle than the length, and so E is unifom. E fo paallel plates is Eσ/ε, σ is the suface chage density. b V ba E dl a E and V ae elated Since we take the integal fom lowe potential point a to highe potential point b along the field line, we obtain V ba Vb Va b Edl cos18 + b Edl b σ b Q Q b a a dl a ε dl dl a ε A ε A a ε A So fom the fomula: Q Q ε A What do you notice? C V ba Qd ε A d Q b a ( ) C only depends on the aea and the distance of the plates and the pemittivity of the medium between them. 16 Qd ε A

17 Example 24 1 Capacito calculations: (a) Calculate the capacitance of a capacito whose plates ae 2cmx3.cm and ae sepaated by a 1.mm ai gap. (b) What is the chage on each plate if the capacito is connected to a 12-V battey? (c) What is the electic field between the plates? (d) Estimate the aea of the plates needed to achieve a capacitance of 1F, given the same ai gap. (a) Using the fomula fo a paallel plate capacito, we obtain ε C d A m ( ) C N m 53 1 C N m 53pF 1 1 m (b) Fom QCV, the chage on each plate is Q C N m 12V C 64pC CV ( )( ) 17

18 Example 24 1 (C) Using the fomula fo the electic field in two paallel plates σ E ε Q Aε C 6. 1 m C N m O, since V Ed we can obtain E (d) Solving the capacitance fomula fo A, we obtain ε C A d Solve fo A NC Vm V d 12V Vm m A Cd ε 1F m ( C N m ) m 1km About 4% the aea of Alington (256km 2 ). 18

19 Example 24 3 Spheical capacito: A spheical capacito consists of two thin concentic spheical conducting shells, of adius a and b, as in the figue. The inne shell caies a unifomly distibuted chage Q on its suface and the oute shell and equal but opposite chage Q. Detemine the capacitance of the two shells. Using Gauss law, the electic field outside a unifomly chaged conducting sphee is So the potential diffeence between a and b is b Vba E dl a b b b Q Q b d Q 1 d Q a 2 E d 2 a Thus capacitance is a a Q Q C V Q a b b a E b a Q Q b a a b a b a 19 b

PHYS 1444 Lecture #5

PHYS 1444 Lecture #5 Shot eview Chapte 24 PHYS 1444 Lectue #5 Tuesday June 19, 212 D. Andew Bandt Capacitos and Capacitance 1 Coulom s Law The Fomula QQ Q Q F 1 2 1 2 Fomula 2 2 F k A vecto quantity. Newtons Diection of electic