Capacitance and Dielectrics

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3/30/05 apacitance an Dielectrics Goals of this

capacitance To analyze networks of capacitors

Lawal Department of Physics University of

n Introuction Introuction How o camera flash

1 3/30/05 apacitance an Dielectrics Goals of this Lecture To unerstan capacitors an calculate capacitance To analyze networks of capacitors To calculate the enery store in a capacitor To examine ielectrics an how they affect capacitance T. O. Lawal Department of Physics University of Ilorin Lawal.to@unilorin.eu.n Introuction Introuction How o camera flash units store enery? apacitors are evices that store electric potential enery. Enery of capacitor is store in E fiel.

capacity hare a capacitor by pushin it there with a potential voltae pressure apacitors an apacitance A capacitor in a simple electric circuit.

2 3/30/05 Introuction apacitors an capacitance Any two conuctors separate by an insulator form a capacitor. Insulator will allow E fiel between the conuctors, Insulator will not allow chare to flow from one conuctor throuh itself to the other. The more chare you can hol, the larer the capacitor! capacity hare a capacitor by pushin it there with a potential voltae pressure apacitors an apacitance A capacitor in a simple electric circuit. hare store: V The store chare is proportional to the potential ifference V between the plates. The capacitance is the constant of proportionality, measure in Faras. Fara = oulomb / Volt apacitors an capacitance The efinition of capacitance is = /V ab. = chare store is hel symmetrically (same + as ) V ab = pressure that pushes an keeps chare there increases as increases more capacity! ecreases as V ab increases more pressure require to hol the chare there, so less effective in storin it temporarily!

3/30/05 Forms of apacitors Parallel-plate capacitor TWO parallel conuctin plates Separate by

Parallel-Plate apacitor ylinrical apacitor A cylinrical capacitor is a parallel-plate capacitor

Parallel-plate capacitor The capacitance of a parallel-plate capacitor is = e 0 A/. Note!

increases with Area ecreases with separation Parallel-plate capacitor Plates.00 m in area; 5.

3 3/30/05 Forms of apacitors Parallel-plate capacitor TWO parallel conuctin plates Separate by istance that is small compare to their imensions. Parallel-Plate apacitor ylinrical apacitor A cylinrical capacitor is a parallel-plate capacitor that has been rolle up with an insulatin layer between the plates. Parallel-plate capacitor The capacitance of a parallel-plate capacitor is = e 0 A/. Note! is enineere! You control Area & istance by esin! increases with Area ecreases with separation Parallel-plate capacitor Plates.00 m in area; 5.00 mm apart; 0 kv applie Potential Difference. = e 0 A/ = 8.85 x 0 - F/m *.00 m /.005 m = 3.54 x 0-9 F on each plate from = /V so = V = 3.54 x 0-5 E fiel between plates from V = E so E = V/ =.00 x 0-6 V/m Or E = s/e 0 = (/Area)/e 0 = 3.54 x 0-5 /.00 m / e 0 3

4 3/30/05 A spherical capacitor Two concentric spherical shells separate by vacuum.. What is? A spherical capacitor Two concentric spherical shells separate by vacuum.. What is? = /V so we nee V! Get V from E fiel! Er = /4pe 0 / r V a V b = - interal (E.l) V = kq/r in eneral for spherical chare istribution V ab = V a V b = k/r a k/r b = k (/r a /r b ) V ab = /4pe 0 (/r a /r b ) = /V = 4pe 0 (/r a /r b ) = 4pe 0 (r a r b )/ (r b r a ) As r b tens to infinity, i.e, r b r a tens r b then = 4pe 0 r a A cylinrical capacitor Linear chare ensity +l on outer cyliner of raius r b, - l on inner cyliner of raius r a. What is? = /V ab ; fin an fin V ab! = l L V = [l/pe 0 ] ln (r 0 /r) r 0 = istance where V was efine to be zero! Say r 0 = r b here, so V ab = (l/pe 0 ln (r b /r a ) A cylinrical capacitor Linear chare ensity +l on outer cyliner of raius r b, - l on inner cyliner of raius r a. What is? = /Vab = l L/[(l/pe 0 ln (r b /r a )] = pe 0 L/ ln (r b /r a ) heck: Units = Faras/Meter x Meters = Faras increases as L increases, an increaes as r b closer to r a! 4

5 3/30/05 apacitors in series apacitors are in series if connecte one after the other We also have to have that the potential across plus the potential across shoul equal the potential rop across the two capacitors Vab Vac Vcb V V We have Then apacitors in Series V V ab anv Diviin throuh by, we have V ab V ab apacitors in Series The equivalent capacitor will also have the same voltae across it The left han sie is the inverse of the efinition of capacitance So we then have for the equivalent capacitance eq If there are more than two capacitors in series, the resultant capacitance is iven by eq i i V apacitors in parallel apacitors are connecte in parallel between a an b if potential ifference V ab is the same for all the capacitors. 5

6 3/30/05 We have that Now apacitors in Parallel or The upper plates of both capacitors are at the same potential V V V V an Likewise for the bottom plates ab V V an V apacitors in Parallel The equivalent capacitor will have the same voltae across it, as o the capacitors in parallel But what about the chare on the equivalent capacitor? The equivalent capacitor will have the same total chare Usin this we then have eq or V V V eq apacitors in Parallel The equivalent capacitance is just the sum of the two capacitors a b 3 Example a b If we have more than two, the resultant capacitance is just the sum of the iniviual capacitances eq i i Where o we start? econize that an are parallel with each other an combine these to et This is then in series with with 3 The resultant capacitance is then iven by 3 3 ( ) 3 6

7 This imae cannot currently be isplaye. 3/30/05 onfiuration A onfiuration B onfiuration Three confiurations are constructe usin ientical capacitors Which of these confiurations has the lowest overall capacitance? a) onfiuration A b) onfiuration B c) onfiuration Example The net capacitance for A is just In B, the caps are in series an the resultant is iven by In, the caps are in parallel an the resultant is iven by net net net A circuit consists of three unequal capacitors,, an 3 which are connecte to a battery of emf E. The capacitors obtain chares, 3, an have voltaes across their plates V, V, an V 3. eq is the equivalent capacitance of the circuit. heck all of the followin that apply: Example 3 a) = b) = 3 c) V = V 3 ) E = V e) V < V f) eq > A etaile worksheet is available etailin the answers What is the equivalent capacitance, eq, of the combination shown? Example 4 o eq o alculations of capacitance Example 4.6, a capacitor network: Fin eq? (a) eq = (3/) (b) eq = (/3) (c) eq = 3 3 eq 7

8 3/30/05 alculations of capacitance Example 4.6, a capacitor network: Fin eq? alculations of capacitance Example 4.6, a capacitor network: Fin eq? Enery Store in a apacitor Electrical Potential enery is store in a capacitor The enery comes from the work that is one in charin the capacitor Let q an v be the intermeiate chare an potential on the capacitor The incremental work one in brinin an incremental chare, q, to the capacitor is then iven by W v q q q Enery Store in a apacitor The total work one is just the interal of this equation from 0 to W 0 q q Usin the relationship between capacitance, voltae an chare we also obtain U V V where U is the store potential enery 8

9 3/30/05 Example 5 Suppose the capacitor shown here is chare to an then the battery is isconnecte Now suppose you pull the plates further apart so that the final separation is Which of the quantities,, V, U, E chane? : : V: U: How o these quantities chane? Answers: hare on the capacitor oes not chane apacitance Decreases Voltae Increases Potential Enery Increases E: Electric Fiel oes not chane V V A U U Suppose the battery (V) is kept attache to the capacitor Aain pull the plates apart from to Now which quantities, if any, chane? : : V: U: E: How much o these quantities chane? Answers: Example 6 hare Decreases apacitance Decreases Voltae on capacitor oes not chane Potential Enery Decreases Electric Fiel Decreases U E E U Electric Fiel Enery Density The potential enery that is store in the capacitor can be thouht of as bein store in the electric fiel that is in the reion between the two plates of the capacitor The quantity that is of interest is in fact the enery ensity Enery Density u V where A an are the area of the capacitor plates an their separation, respectively A Usin Electric Fiel Enery Density A e0 an V E u e 0 E we then have Even thouh we use the relationship for a parallel capacitor, this result hols for all capacitors rearless of confiuration This represents the enery ensity of the electric fiel in eneral 9

10 3/30/05 Dielectrics A ielectric is an insulatin material (e.. paper, plastic, lass). A ielectric place between the conuctors of a capacitor increases its capacitance by a factor κ, calle the ielectric constant. = κ o ( o =capacitance without ielectric) e 0 A A e For a parallel-plate capacitor: ε = κε o = permittivity of the material. Properties of Dielectric Materials Dielectric strenth is the maximum electric fiel that a ielectric can withstan without becomin a conuctor. Dielectric materials increase capacitance. increase electric breakown potential of capacitors. provie mechanical support. Material Dielectric onstant κ Dielectric Strenth (V/m) air x 0 6 paper x 0 6 mica 7 50 x 0 6 strontium titanate x 0 6 harin a capacitor harin a capacitor At time t=0 the switch is close, with the capacitor initially unchare. A current will flow e=v c +V =I 0, as initially V c =0. Thus the initial current is I 0 =e/. Now a chare beins to buil on the capacitor, introucin a reverse voltae. The current falls, an stops when the P.D. across is e. Final chare is iven by "=V" => 0 =e. 0

11 3/30/05 harin a capacitor (quantitative). Apply Kirchoff's loop rule. e t V V c e i e t e ln( e ) t t t 0 0 ln( e ) ln( e ) t t harin a capacitor (cont) e ln e t e t / e e t / e 0 Where 0 = e = the final chare on the capacitor. ( e 0 t / ) harin a capacitor (cont). To fin the current, ifferentiate since I=/t. ( e 0 I e e e 0 t / t / By consierin time zero, when the current is I 0, I I e t / ). t / 0

12 3/30/05 Introuction Brie are use to measure resistance, inuctance, capacitance an impeance. Their operation is base on a null inication principle. This means the inication is inepenent of the calibration of the inicatin evice or any characteristics of it. Very hih erees of accuracy can be achieve usin the bries Incientally, raphs can be V or aainst t, they all have the same basic shape Types of bries Two types of brie are use in measurement: ) D brie: a) Wheatstone Brie b) Kelvin Brie ) A brie: a) Similar Anle Brie b) Opposite Anle Brie/Hay Brie c) Maxwell Brie ) Wein Brie e) aio Frequency Brie f) Scherin Brie D BIDGES The Wheatstone Brie The Kelvin Brie

3/30/05 Wheatstone Brie A Wheatstone brie is a measurin instrument invente by Samuel Hunter hristie (British scientist & mathematician) in 833 an improve an popularize by Sir harles Wheatstone in 843.

Its operation is similar to the oriinal potentiometer except that in potentiometer circuits the meter use is a sensitive alvanometer.

13 3/30/05 Wheatstone Brie A Wheatstone brie is a measurin instrument invente by Samuel Hunter hristie (British scientist & mathematician) in 833 an improve an popularize by Sir harles Wheatstone in 843. It is use to measure an unknown electrical resistance by balancin two les of a brie circuit, one le of which inclues the unknown component. Its operation is similar to the oriinal potentiometer except that in potentiometer circuits the meter use is a sensitive alvanometer. Sir harles Wheatstone (80 875) Wheatstone Brie Definition: Basic circuit confiuration consists of two parallel resistance branches with each branch containin two series elements (resistors). To measure instruments or control instruments Basic c brie use for accurate measurement of resistance: Fi. 5.: Wheatstone brie circuit How a Wheatstone Brie works? The c source, E is connecte across the resistance network to provie a source of current throuh the resistance network. The sensitive current inicatin meter or null etector usually a alvanometer is connecte between the parallel branches to etect a conition of balance. When there is no current throuh the meter, the alvanometer pointer rests at 0 (miscale). urrent in one irection causes the pointer to eflect on one sie an current in the opposite irection to otherwise. The brie is balance when there is no current throuh the alvanometer or the potential across the alvanometer is zero. ont. At balance conition; voltae across an also equal, therefore () I I Voltae rop across 3 an 4 is equal I 3 3 = I 4 4 () No current flows throuh alvanometer G when the brie is balance, therefore: I = I 3 an I =I 4 (3) 3

14 3/30/05 ont. Substitute (3) in Eq (), I 3 = I 4 (4) Eq (4) evie Eq () Example 5- Fiure 5. consists of the followin, = k, = 5 k, 3 = 3 k. Fin the unknown resistance x. Assume a null exists(current throuh the alvanometer is zero). / 3 = / 4 Then rewritten as 4 = 3 (5) Fi. 5-: ircuit For example 5- Solution 5- ont. x = 3 x = 3 / = (5 x 3)/ k, = 40 k x 4

15 3/30/05 ont. X is the unknown esistance, P,, are resistance boxes. One of the resistance box (e.. ) is ajuste until the alvanometer between A,, represente by its resistance, shows no eflection that is I = 0 Then P/ = /X X = /P Usin Kirchorff s law applie to the cct. Loop ABA: I I PI 0 Loop ADA: I X( I I ) ( I I ) 0 I I X I X I I 0 I ( X ) XI I 0 ont. Suppose we wish to fin the conition for no eflection on the alvanometer, then we put I = 0 in equ () an (), we have An I PI 0 or PI I P I I Therefore. This is the conition for balance of the brie. IX I 0 o r X I I X I I P X ont. ont. We want to look at how the alvanometer in a brie circuit can be positione. This is an unbalance Wheatstone brie where I is to be etermine. The internal resistance r is neliible. Applyin Kirchoff s law Loop ABA: Loop ADA: 0I 0( I I ) 00( I I ) 0 30I 0I 00I 0 q I 0I 3 I * I I I

16 3/30/05 Subs for ont. I 99I 0(0 I 3 I ) 0I 0 99I 00I 60I 0I 0 I I 80I 0 80I Subs for I in equ (**). ont I 0I I.5 Subs for I in equ (*) I 800I 3I 787I I Loop DBXD: 0I 0( I I ).5 30I 0I.5 (**) A 5 A Sensitivity of the Wheatstone Brie When the brie is in unbalance conition, current flows throuh the alvanometer, causin a eflection of its pointer. The amount of eflection is a function of the sensitivity of the alvanometer. ont. Deflection may be expresse in linear or anular units of measure, an sensitivity can be expresse: milimeters erees raians S A A A Total eflection, D S I 6

3/30/05 METE BIDGE A meter brie is a simple an cheap form of a Wheatstone brie. It is sometimes calle a slie-wire brie. METE BIDGE If x is to be etermine from cct.

17 3/30/05 METE BIDGE A meter brie is a simple an cheap form of a Wheatstone brie. It is sometimes calle a slie-wire brie. METE BIDGE If x is to be etermine from cct. Shown in the previous slies, then we assume that the slie wire is uniform. At null conition, (I B A) an (I B K A) Therefore, applyin Loops BKB = I I 3 or I I 3 0 an AKA I I x hence x 3 METE BIDGE But if L A L an A Therefore, L L L x 3 L If Then METE BIDGE an lenth A 8. cm, B 7.8cm x 8. (34.5) Example. In a meter brie cct. A balance (zero eflection in the alvanometer) is obtaine at 0.6 m mark. When a 6 ohm resistor is connecte in the left ap. al. The value of the resistance in the riht ap of the cct

18 3/30/05 METE BIDGE From the cct. Shown below, the resistance of X an Y are 5 Ohms an 3 Ohms resp. the lenth of AB =.00m. When shunt is connecte in parallel to X, the balance lenth is 0.57m from A. what is the resistance of the shunt. POTENTIOMETE POTENTIOMETE A potentiometer is another null-type instrument which is uner zero current conition, to measure the potential ifferences by comparison with a stanar voltae source. The workin battery supplies current I to wire AB. When the switch is then thrown to position () an the unknown voltae Ex is connecte. Point is move alon AB until no current flows throuh G. Therefore, Loop AKA Ix Ex 0 x x If the switch is now thrown to position (), Es is connecte. The pointer is move alon AB to a new position s is the resistance between A an So Loop A LA : I E I E 0 I s s E s s b a POTENTIOMETE Diviin equ (a) by equ (b) E I E I E x x x s s s x x E s s = E = E s s L L x s A A ' 8

19 3/30/05 POTENTIOMETE Ex. A slie-wire potentiometer is balance aainst a.08v stanar cell when the slie wire is set at 40.0 cm out of a total lenth of 00 cm. for an unknown source, the settin is.9 cm. what is the emf of the unknown cell. Solution. x Lx Ex Es Es s Ls.9 = (.08) 40. = V 9

UNIT 4:Capacitors and Dielectric

UNIT 4:Capacitors and Dielectric UNIT 4:apacitors an Dielectric SF7 4. apacitor A capacitor is a evice that is capable of storing electric charges or electric potential energy. It is consist of two conucting plates separate by a small