PHYS 1444 Lecture #5

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1 Shot eview Chapte 24 PHYS 1444 Lectue #5 Tuesday June 19, 212 D. Andew Bandt Capacitos and Capacitance 1

2 Coulom s Law The Fomula QQ Q Q F Fomula 2 2 F k A vecto quantity. Newtons Diection of electic (Coulom) foce (Newtons) is always along the line joining the two ojects. Unit of chage is called Coulom, C, in SI. Elementay chage, the smallest chage, is that of an electon: -e whee 19 e C 2

3 Vecto Polems Calculate magnitude of vectos (Ex. foce using Coulom s Law) Split vectos into x and y components and add these sepaately, using diagam to help detemine sign Calculate magnitude of esultant F = (F x2 +F y2 ) Use = tan -1 (F y /F x ) to get angle 3

4 Gauss Law The pecise elation etween flux and the enclosed chage is given y Gauss Law E da Q encl is the pemittivity of fee space in the Coulom s law A few impotant points on Gauss Law Feedom to choose suface Distiution of chages inside suface does not matte only total chage Chages outside the suface do not contiute to Q encl. 4

5 Example 22-3: Spheical conducto. A thin spheical shell of adius possesses a total net chage Q that is unifomly distiuted on it. Detemine the electic field at points (a) outside the shell, and () within the shell. (c) What if the conducto wee a solid sphee? *q3 E da Q encl Figue Coss-sectional dawing of a thin spheical shell of adius caying a net chage Q unifomly distiuted. A 1 and A 2 epesent two gaussian sufaces we use to detemine Example Solution: a. The gaussian suface A 1, outside the shell, encloses the chage Q. We know the field must e adial, so E = Q/(4πε 2 ).. The gaussian suface A 2, inside the shell, encloses no chage; theefoe the field must e zeo. c. All the excess chage on a conducto esides on its suface, so these answes hold fo a solid sphee as well. Tuesday, June 19, 212 Key to these questions is how much chage is enclosed PHYS 1444 D. Andew Bandt 5

6 Example 22-4: Solid sphee of chage. An electic chage Q is distiuted unifomly thoughout a nonconducting sphee of adius. Detemine the electic field (a) outside the sphee ( > ) and () inside the sphee ( < ). *q4 Solution: a. Outside the sphee, a gaussian suface encloses the total chage Q. Theefoe, E = Q/(4πε 2 ).. Within the sphee, a spheical gaussian suface encloses a faction of the chage Q 3 / 3 (the atio of the volumes, as the chage density is constant). Integating and solving fo the field gives E = Q/(4πε 3 ). 6

7 Example 22-5: Nonunifomly chaged solid sphee. Suppose the chage density of a solid sphee is given y ρ E = α 2, whee α is a constant. (a) Find α in tems of the total chage Q on the sphee and its adius. () Find the electic field as a function of inside the sphee. Solution: a. Conside the sphee to e made of a seies of spheical shells, each of adius and thickness d. The volume of each is dv = 4π 2 d. To find the total chage: Q = ρ E dv = 4πα 5 /5, giving α = 5Q/4π 5.. The chage enclosed in a sphee of adius will e Q 5 / 5. Gauss s law then gives E = Q 3 /4πε 5. 7

8 Electic Potential Enegy Concept of enegy is vey useful solving mechanical polems Consevation of enegy makes solving complex polems easie. Defined fo consevative foces (independent of path) 8

9 Popeties of the Electic Potential What ae the diffeences etween the electic potential and the electic field? 1 Q V Electic potential (U/q) Simply add the potential fom each of the chages to otain the total potential fom multiple chages, since potential is a scala quantity Electic field (F/q) 1 4 Need vecto sums to otain the total field fom multiple chages Potential fo a positive chage is lage nea a positive chage and deceases to at lage distances. Potential fo the negative chage is small (lage magnitude ut negative) nea the chage and inceases with distance to E 4 Q 2 9

10 Capacitos (o Condenses) What is a capacito? A device that can stoe electic chage without letting the chage flow What does it consist of? Usually consists of two oppositely chaged conducting ojects (plates o sheets) placed nea each othe without touching Why can t they touch each othe? The chages will neutalize each othe Can you give some examples? Camea flash, suge potectos, compute keyoad, inay cicuits How is a capacito diffeent than a attey? Battey povides potential diffeence y stoing enegy (usually chemical enegy) while the capacito stoes chage ut vey little enegy. 1

11 Capacitos A simple capacito consists of a pai of paallel plates of aea A sepaated y a distance d. A cylindical capacitos ae essentially paallel plates wapped aound as a cylinde. Symols fo a capacito and a attey: Capacito - - Battey (+) - i- (-) Cicuit Diagam Tuesday, June 19, 212 PHYS 1444 D. Andew Bandt 11

12 Capacitos What do you think will happen if a attey is connected (voltage is applied) to a capacito? The capacito gets chaged quickly, one plate positive and the othe negative with an equal amount. of chage Each attey teminal, the wies and the plates ae conductos. What does this mean? All conductos ae at the same potential. the full attey 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 a etween the plates. How would you wite this fomula? Q CV C is a popety of a capacito so does not depend on Q o V. a C is a popotionality constant, called capacitance of the device. What is the unit? C/V o Faad (F) Nomally use F o pf. 12

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

13 Detemination of Capacitance C can e detemined analytically fo capacitos w/ simple geomety and ai in etween. Let s conside a paallel plate capacito. Plates have aea A each and sepaated y d. d is smalle than the length, so E is unifom. Fo paallel plates E=, whee is the suface chage density. E and V ae elated V E dl a a Since we take the integal fom the lowe potential point a to the highe potential point along the field line, we otain V a V V Edl cos18 a a So fom the fomula: Edl a a a a Q Q A C C only depends on the aea V Qd A d (A) and the sepaation (d) of What do you notice? a the plates and the pemittivity of the medium etween 13 them. dl Q dl Q dl Q a Qd A A A A

14 Example 24 1 Capacito calculations: (a) Calculate the capacitance of a capacito whose plates ae 2 cm x 3. cm and ae sepaated y a 1. mm ai gap. () What is the chage on each plate if the capacito is connected to a 12 V attey? (c) What is the electic field etween 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 otain C A d m C N m 53 1 C N m 53pF 1 1 m () Fom Q=CV, the chage on each plate is Q CV C N m 12V C 64 pc 14

15 E Example 24 1 (c) Using the fomula fo the electic field in two paallel plates O, since V Ed we can otain E (d) Solving the capacitance fomula fo A, we otain C Q A A d Solve fo A m C N m C V d 12V N C m V m V m A Cd 1F C m N m m 1km Aout 4% the aea of Alington (256km 2 ). 15

Detemine the capacitance of this configuation.

16 Example 24 3 Spheical capacito: A spheical capacito consists of two thin concentic spheical conducting shells, of adius a and, as in the figue. The inne shell caies a unifomly distiuted chage Q on its suface and the oute shell an equal ut opposite chage Q. Detemine the capacitance of this configuation. Using Gauss law, the electic field outside a unifomly chaged conducting sphee is So the potential diffeence etween a and is V a a E dl E d Q 2 d a a 4 Thus capacitance is 2 a 4 a C Q d Q 4 4 Q Q V Q a 4 a 1 a Q E 4 a a 4 Q Q 4 a a 16

17 Capacito Cont d A single isolated conducto can e said to have a capacitance, C. C can still e defined as the atio of the chage to asolute potential V on the conducto. So Q=CV. The potential of a single conducting sphee of adius can e otained as V Q 4 a 1 1 Q 4 whee So its capacitance is C Q / V 4 Although it has capacitance, a single conducto is not consideed to e a capacito, as a second neay chage is equied to stoe chage a Tuesday, June 19, 212 PHYS 1444 D. Andew Bandt 17

18 Capacitos in Seies o Paallel Capacitos ae used in many electic cicuits What is an electic cicuit? A closed path of conductos, usually wies connecting capacitos and othe electical devices, in which chages can flow thee is a voltage souce such as a attey Capacitos can e connected in vaious ways. In paallel and in Seies o in comination 18

19 Capacitos in Paallel Paallel aangement povides the same voltage acoss all the capacitos. Left hand plates ae at V a and ight hand plates ae at V So each capacito plate acquies chages given y the fomula Q 1 =C 1 V, Q 2 =C 2 V, and Q 3 =C 3 V The total chage Q that must leave attey is then Q=Q 1 +Q 2 +Q 3 =V(C 1 +C 2 +C 3 ) Conside that the thee capacitos ehave like a single equivalent one Q=C eq V= V(C 1 +C 2 +C 3 ) Fo capacitos in paallel the capacitance is the sum of the individual capacitos Thus the equivalent capacitance in paallel is C eq C 1 C 2 19 C 3

20 Capacitos in Seies Seies aangement is moe inteesting When attey is connected, +Q flows to the left plate of C 1 and Q flows to the ight plate of C 3 This induces opposite sign chages on the othe plates. Since the capacito in the middle is oiginally neutal, chages get induced to neutalize the induced chages So the chage on each capacito is the same value, Q. (Same chage) Conside that the thee capacitos ehave like an equivalent one Q=C eq V V=Q/C eq The total voltage V acoss the thee capacitos in seies must e equal to the sum of the voltages acoss each capacito. V=V 1 +V 2 +V 3 =(Q/C 1 +Q/C 2 +Q/C 3 ) Putting all these togethe, we otain: V=Q/C eq =Q(1/C 1 +1/C 2 +1/C 3 ) Thus the equivalent capacitance is Ceq C1 C2 C3 PHYS 1444 D. Andew Bandt The total capacitance is smalle than the smallest C!!! 2

PHYS 1444 Section 501 Lecture #7

PHYS 1444 Section 501 Lecture #7 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