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1 Get Solution of These Packages & Learn by Vieo Tutorials on FREE Downloa Stuy Package from website: & INTRODUTION APAIT ITANE A capacitor can store energy in the form of potential energy in an electric fiel. In this chapter we'll iscuss the capacity of conuctors to hol charge an energy.. APAIT ITANE OF AN ISOLATED ONDUTOR OR When a conuctor is charge its potential increases. It is foun that for an isolate conuctor (conuctor shoul be of finite imension, so that potential of infinity can be assume to be zero). Potential of the conuctor is proportional to charge given to it. q charge on conuctor V potential of conuctor q V q V Where is proportionality constant calle capacitance of the conuctor.. Definition of capacitance : apacitance of conuctor is efine as charge require to increase the potential of conuctor by one unit.. Important points about the capacitance of an isolate conuctor : (i) It is a scalar quantity. (ii) Unit of capacitance is fara in SI units an its imensional formula is M L Ι T 4 (iii) Fara : Fara is the capacitance of a conuctor for which coulomb charge increases potential by volt. Fara oulomb Volt µf 6 F, nf 9 F or pf F (iv) apacitance of an isolate conuctor epens on following factors : (v) (a) Shape an size of the conuctor : On increasing the size, capacitance increases. (b) On surrouning meium : With increase in ielectric constant K, capacitance increases. (c) Presence of other conuctors : When a neutral conuctor is place near a charge conuctor capacitance of conuctors increases. apacitance of a conuctor o not epen on (a) harge on the conuctor (b) Potential of the conuctor (c) Potential energy of the conuctor.. POTENTIAL ENERGY OR SELF ENERGY OF AN ISOLATED ONDUTOR OR Work one in charging the conuctor to the charge on it against its own electric fiel or total energy store in electric fiel of conuctor is calle self energy or self potential energy of conuctor.. Electric potential energy (Self Energy) : U q V qv q harge on the conuctor V Potential of the conuctor Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

2 FREE Downloa Stuy Package from website: & Get Solution of These Packages & Learn by Vieo Tutorials on Ex. apacitance of the conuctor.. Self energy is store in the electric fiel of the conuctor with energy ensity (Energy per unit volume) U E V [The energy ensity in a meium is r E ] where E is the electric fiel at that point.. In case of charge conuctor energy store is only out sie the conuctor but in case of charge insulating material it is outsie as well as insie the insulator. (i) When coulomb charge is assigne to an isolate conuctor its potential becomes 5 volt, fin out capacitance of the conuctor? (ii) If now further coulomb charge is supplie to it then what is the new potential on conuctor? (i) V Q Ex. (ii) V Fara. 5 Q 5 volt. An isolate conuctor of µf capacitance is given µ charge. Fin out store energy an its potential? Store energy U V Q. µ. (µ ). µ F.5 µj Q µ Potential V volt. µ F Q. Potential of an isolate conuctor is foun to become volt when a charge µ is given to it, answer the following : (i) What is the capacitance of conuctor? (ii) (iii) If now further 4 µ charge is ae to it then fin out potential of conuctor now? What is the energy store in two cases? Ans. (i) µf (ii) 6 V (iii) U i µj, U f 8 µj. APAIT ITANE OF AN ISOLATED SPHERIAL ONDUTOR OR (i) (ii) If the meium aroun the conuctor is vacuum or air. Vacuum 4π R R Raius of spherical conuctor. (may be soli or hollow.) If the meium aroun the conuctor is a ielectric of constant K from surface of sphere to infinity. Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

3 Get Solution of These Packages & Learn by Vieo Tutorials on meium 4π KR FREE Downloa Stuy Package from website: & Ex. (iii) meium air / vaccum K ielectric constant. 8 similar charge rops combine to from a bigger rop. The ratio of the capacity of bigger rop to that of smaller rop will be- (A) : (B) 8 : () 4 : (D) 6 : Sol: bigger rop ( small rop )n /...() n 8...() begger rop small rop Hence the correct answer will be (A). 4. SHARING OF HARGES ON JOINING TWO T HARGED ONDUTORS ORS (i) (ii) (iii) (iv) (v) (vi) Whenever there is potential ifference there will be flow of charge. harge always have tenency to flows from high potential energy to low potential energy when release freely. Positive charge always flows from high potential to low potential [if only electric force act on charge]. Negative charge always flows from low potential to high potential [if only electric force act on charge]. The flow of charge will continue till there is potential ifference between the conuctors (finally potential ifference ). Formulae relate with reistribution of charges. Before connecting the conuctors Parameter onuctor I st onuctor II n apacitance harge Q Q Potential V V After connecting the conuctors Parameter I st onuctor II n onuctor apacitance V harge ' Q Potential V V ' Q Q ' ' ' Q Q ' Q Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

4 Get Solution of These Packages & Learn by Vieo Tutorials on But, ' Q Q ' Q Q FREE Downloa Stuy Package from website: & Ex. 4 V Q Q Q ' & Q ' V V (Q Q ) (Q Q ) Heat loss uring reistribution : H (V V ) The loss of energy is in the form of Joule heating in the wire. Note : Always put Q, Q, V an V with sign. A an B are two isolate conuctors (that means they are place at a large istance from each other). When they are joine by a conucting wire: (i) Fin out final charges on A an B? (ii) (iii) Fin out heat prouce uring the process of flow of charges. Fin out common potential after joining the conuctors by conucting wires? (i) Q A ' (6 ) µ 6 6 Q B ' (6 ) 6µ 6 µ F.6µ F (ii) h.. (µ F 6µ F). (µf). µj µ 6µ (iii) V volt. µ F 6µ F Q. When two isolate conuctors A an B are connecte by a conucting wire positive charge will flow from. Ans. B. (A) A to B (B) B to A () will not flow (D) can not say. Q. A conuctor of capacitance µf connecte to other conuctor of capacitance 4 µf having equal charges µ initially. Fin out final voltage an heat loss uring the process? Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 4 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

5 Get Solution of These Packages & Learn by Vieo Tutorials on Ans. (i) V 4V (ii) H 5 µj. FREE Downloa Stuy Package from website: & 5. APAIT ITOR : A capacitor or conenser consists of two conuctors separate by an insulator or ielectric. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) When uncharge conuctor is brought near to a charge conuctor, the charge on conuctors remains same but its potential ecreases resulting in the increase of capacitance. In capacitor two conuctors have equal but opposite charges. The conuctors are calle the plates of the capacitor. The name of the capacitor epens on the shape of the capacitor. Formulae relate with capacitors (a) (b) Q V Q QA V V V A B QB V V Q harge of positive plate of capacitor. B A V Potential ifference between positive an negative plates of capacitor apacitance of capacitor. Energy store in the capacitor U V Q QV This energy is store insie the capacitor in its electric fiel with energy ensity U E V or r E. The capacitor is represente as following:, Base on shape an arrangement of capacitor plates there are various types of capacitors. (a) (b) (c) Parallel plate capacitor. Spherical capacitor. ylinrical capacitor. apacitance of a capacitor epens (a) (b) (c) Area of plates. Distance between the plates. Dielectric meium between the plates. apacitance of a parallel plate capacitor (air fille) is given by following formula where A A area of the plates. istance between plates. (ix) Electric fiel intensity between the plates of capacitors (air fille ) (x) E σ/ V/ Force experience by any plate of capacitor F q /A 6. IRUIT UIT SOLUTION FOR F R IRUIT UIT AT t (INITIAL STATE) TE) Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 5 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

6 FREE Downloa Stuy Package from website: & Get Solution of These Packages & Learn by Vieo Tutorials on AND AT t (FINAL STATE) TE) Note : (i) Ex.4 (ii) (iii) U Ex.5 Ex.6 harge on the capacitor oes not change instantaneously or suenly if there if a resistance in the path (series) of the capacitor. When an uncharge capacitor is connecte with battery then its charge is zero initially hence potential ifference across it is zero initially. At this time the capacitor can be treate as a conucting wire The current will become zero finally (that means in steay state) in the branch which contains capacitor. A capacitor of capacity µf is charge to a potential ifference of KV. The energy store in the capacitor will be- (A).5 joule (B) joule ().5 erg. (D) erg. V 6.5 Joule. Fin out current in the circuit an charge on capacitor which is initially uncharge in the following situations. (a) Just after the switch is close. (b) After a long time when switch was close. For just after closing the switch: potential ifference across capacitor Q i 5A After a long time at steay state current i an potential ifference across capacitor V Q q Fin out current i, i, i, charge on capacitor an of capacitor in the t circuit which is initially uncharge in the following situations. (a) Just after the switch is close (b) After a long time when switch is close. Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 6 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

7 Get Solution of These Packages & Learn by Vieo Tutorials on Initially the capacitor is uncharge so its behaviour is like a conuctor FREE Downloa Stuy Package from website: & Ex.7 Let potential at A is zero so at B an also zero an at F it is. Let potential at E is x so at D also x. Apply Kirchhoff s Ι st law at point E : x R x R x R x R / Ι R Ι Ι Alternatively i R R R eq i i i at t (finally) R R R x R Q c R R capacitor completely charge so their will be no current through it. Q Ι, Ι Ι R V e V B V D V (/R)R / Time t Finally t, Q Ι t Ι Ι Ι Q Q /t R R R R R R At t switch S is close an remains close for a long time an S remains open. Now S is opene an S is close assuming that capacitor is initially uncharge fin out (i) The current through the capacitor immeiately after that moment (ii) harge on the capacitor long after that moment. (iii) Total charge flown through the cell of emf after S is close. (i) Let Potential at point A is zero. Then at point B an it will be (because current through the circuit is zero). V B V A ( ) Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 7 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

8 FREE Downloa Stuy Package from website: & Get Solution of These Packages & Learn by Vieo Tutorials on Ex.8 Ex.9 harge on capacitor ( ) (ii) Now S is close an S is open. (p.. across capacitor an charge on it will not change suenly) Potential at A is zero so at D it is. ( ) current through the capacitor (B to D) R R (iii) after a long time i V B V A V D V A Q ( ) The charge on the lower plate (which is connecte to the battery) changes from to. this charge will come form the battery, charge flown from that cell is ownwar. A capacitor of capacitance which is initially uncharge is connecte with a battery. Fin out heat issipate in the circuit uring the process of charging. Final status Let potential at point A is, so at B also an at an D it is. finally, charge on the capacitor Q U i O U f V work one by battery pt w it it. Q. (Now onwars remember that w.. by battery Q if Q has flown out of the cell from high potential an w.. on battery is Q if Q has flown into the cell through high potential) Heat prouce W (U f U i ). A capacitor of capacitance which is initially charge upto a potential ifference is connecte with a battery of emf such that the positive terminal of battery is connecte with positive plate of capacitor. Fin out heat loss in the circuit uring the process of charging. Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 8 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

9 Get Solution of These Packages & Learn by Vieo Tutorials on FREE Downloa Stuy Package from website: & Ex. Since the initial an final charge on the capacitor is same before an after connection. Here no charge will flow in the circuit so heat loss A capacitor of capacitance which is initially charge upto a potential ifference is connecte with a battery of emf / such that the positive terminal of battery is connecte with positive plate of capacitor. After a long time (i) (ii) (iii) Fin out total charge flow through the battery Fin out total work one by battery Fin out heat loss in the circuit uring the process of charging. Let potential of A is so at B it is. So final charge on capacitor / harge flow through the capacitor (/ ) / So charge is entering into battery. finally, hange in energy of capacitor U final U initial 8 Work one by battery 4 8 Work one by battery hange in energy of capacitor Heat prouce Heat prouce DISTRIBUTION OF HARGES ON ONNETING TWO T HARGED APAIT ITORS: When two capacitors are an are connecte as shown in figure Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 9 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

10 FREE Downloa Stuy Package from website: & Get Solution of These Packages & Learn by Vieo Tutorials on Before connecting the capacitors Parameter apacitor I s t apacitor II n Note : (i) (ii) apacitance harge Q Q Potential V V After connecting the capacitors Parameter I s t apacitor II n apacitor apacitance harge Q Q Potential V V (a) ommon potential : V By charge conservation of plates A an before an after connection. Q Q V V Q Q V V (b) Q ' V (Q Q ) Q ' V (Q Q ) (c) Heat loss uring reistribution : H U i U f Total charge Total capaci tance (V V ) The loss of energy is in the form of Joule heating in the wire. When plates of similar charges are connecte with each other ( with an with ) then put all values (Q, Q, V, V ) with positive sign. When plates of opposite polarity are connecte with each other ( with ) then take charge an potential of one of the plate to be negative. Derivation of above formulae : Let potential of B an D is zero an common potential on capacitors is V, then at A an it will be V V V V V Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

11 Get Solution of These Packages & Learn by Vieo Tutorials on FREE Downloa Stuy Package from website: & V V V H V V ( )V V V V V V (V V ) H (V V ) (V V ) ( ) V V when oppositely charge terminals are connecte then V V V V V H V V (V V ) V V V Ex. Fin out the following if A is connecte with an B is connecte with D. (i) How much charge flows in the circuit. (ii) How much heat is prouce in the circuit. Let potential of B an D is zero an common potential on capacitors is V, then at A an it will be V V V 4 5V 7 V 4 volt harge flow 4 8 µ Now final charges on each plate (ii) Heat prouce () () 5 (4) Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

12 FREE Downloa Stuy Package from website: & Get Solution of These Packages & Learn by Vieo Tutorials on J Note. When capacitor plates are joine then the charge remains conserve. Note. We can also use irect formula of reistribution as given above. Ex. Repeat above question if A is connecte with D an B is connecte with. Let potential of B an is zero an common potential on capacitors is V, then at A an D it will be V V V V Now charge on each plate Heat prouce J Note : here heat prouce is more. Think why? Ex. V A µf capacitor is charge to potential of 5V an then connecte in parallel to another capacitor of capacity µf. If the potential of µf capacitor is Volt then the common potential of two will be - () V (B) V () V (D) 4 V V V V Ex.4 Three capacitors as shown of capacitance µf, µf an µf are charge upto potential ifference V, V an 5 V respectively. If terminal A is connecte with D, is connecte with E an F is connecte with B. Then fin out charge flow in the circuit an fin the final charges on capacitors. Let charge flow is q. Now applying kirchhoff's voltage low ( q ) ( q) q 5 q.5 µ Final charges on plates q Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

13 Get Solution of These Packages & Learn by Vieo Tutorials on Ex.5 In the given circuit fin out the charge on each capacitor. (Initially they are uncharge) FREE Downloa Stuy Package from website: & Let potential at A is, so at D it is V, at F it is V an at point G potential is 5V. Now apply kirchhoff s I st law at point E. ( total charge of all the plates connecte to 'E' must be same as before i.e. ) (x ) (x ) (x 5) Final charges : 5x x 4 Q µf ( 4) 5 µ Q µf ( 4) 6µ Q µf (4 ( 5)) 58 µ 7. OMBINATION OF APAIT ITORS : 7. Series ombination : (i) (ii) (iii) When initially uncharge capacitors are connecte as shown in the combination is calle series combination. All capacitors will have same charge but ifferent potential ifference across then. We can say that V Q V potential across Q charge on positive plate of capacitance of capacitor similarly V (iv) V : V : V Q, V : : Q... We can say that potential ifference across capacitor is inversely proportional to its capacitance in series combination. Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

14 Get Solution of These Packages & Learn by Vieo Tutorials on V FREE Downloa Stuy Package from website: & Note : In series combination the smallest capacitor gets maximum potential. (v) V V V V... V... V... Where V V V V (vi) Equivalent apacitance : Equivalent capacitance of any combination is that capacitance which when connecte in place of the combination stores same charge an energy that of the combination. In series : eq... Note : In series combination equivalent is always less the smallest capacitor of combination. (vii) Energy store in the combination U combination U combination Q Q eq Q Q Energy supplie by the battery in charging the combination U battery Q V Q. U U combination battery Q Q eq eq Note : Half of the energy supplie by the battery is store in form of electrostatic energy an half of the energy is converte into heat through resistance. Derivation of Formulae : meaning of equivalent capacitor Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 4 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

15 Get Solution of These Packages & Learn by Vieo Tutorials on FREE Downloa Stuy Package from website: & Now, eq V Q Initially, the capacitor has no charge. Applying kirchhoff s voltage law in general Q V Q Q Q V. V Q eq eq n n n Ex.6 Three initially uncharge capacitors are connecte in series as shown in circuit with a battery of emf V. Fin out following:- Ex.7 (i) charge flow through the battery, (ii) potential energy in µf capacitor. (iii) U total in capacitors (iv) heat prouce in the circuit 6 6 eq (i) eq µf. Q eq V µ. (ii) charge on µf capacitor µ energy (iii) U total 45 µj Q 5µJ µj (iv) Heat prouce ( µ) () 45 µj 45 µj. Two capacitors of capacitance µf an µf are charge to potential ifference V an 5V as shown in figure. If now terminal B an are connecte together terminal A with positive of battery an D with negative Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 5 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

16 Get Solution of These Packages & Learn by Vieo Tutorials on terminal of battery then fin out final charges on both the capacitor FREE Downloa Stuy Package from website: & Now applying kirchoff voltage law ( q) q 4 q q 6 q harge flow / µ. 7. Parallel ombination : (i) (ii) When one plate of each capacitors (more than one) is connecte together an the other plate of each capacitor is connecte together, such combination is calle parallel combination. All capacitors have same potential ifference but ifferent charges. (iii) We can say that : Q V Q harge on capacitor apacitance of capacitor V Potential of across capacitor (iv) Q : Q : Q : : The charge on the capacitor is proportional to its capacitance (v) Q Q Q Q Q Q Q Where Q Q Q Q... Note : Maximum charge will flow through the capacitor of largest value. (vi) Equivalent capacitance of parallel combination eq Note : Equivalent capacitance is always greater than the largest capacitor of combination. (vii) Energy store in the combination : V combination V V... (...) V eq V U battery QV V Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 6 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

17 Get Solution of These Packages & Learn by Vieo Tutorials on FREE Downloa Stuy Package from website: & U U combination battery Note : Half of the energy supplie by the battery is store in form of electrostatic energy an half of the energy is converte into heat through resistance. Ex.8 Formulae Derivation for parallel combination : In general Q Q Q Q V V V V( ) Q V eq n eq n n Three initially uncharge capacitors are connecte to a battery of V is parallel combination fin out following (i) (ii) (iii) (iv) charge flow from the battery total energy store in the capacitors heat prouce in the circuit potential energy in the µf capacitor. (i) Q ( )µ Ex.9 So. 6 µ (ii) U total 6 µj (iii) heat prouce 6 µj (iv) U µf 5 J In the given circuit fin out charge on 6µF an µf capacitor. It can be simplifie as 8 eq µf 9 charge flow through the cell µ Q 6 µ Now charge on µf harge on 6µF 6 µ Potential ifference across µf 6/ V harge on µf µ. Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 7 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

18 FREE Downloa Stuy Package from website: & Get Solution of These Packages & Learn by Vieo Tutorials on Ex. Ex. Ex. 7. Mixe ombination : The combination which contains mixing of series parallel combinations or other complex combinations fall in mixe category. There are two types of mixe combinations (i) Simple (ii) omplex. 7.4 Simple Mixe ombination : ombinations which can be easily converte in series parallel combination. The given combination is neither a series nor a parallel combination but an are in parallel an that is in series with an 4. Two conensers of same capacity are first connecte in parallel an then in series. The ratio of resultant capacities in two cases will be- (A) : 4 (B) 4 : () : ` (D) : P S P S / 4 Hence the correct answer will be (B). The equivalent capacity in the ajoining figure between the point X an Y will be- (A) 4.5 µf (B) 9µF () µf (D) 6µF Equivalent circuit ( all capacitors are connecte between same p.. x an y ) Ex. All the three conensers are connecte in parallel, hence the resultant capacity 9µF Hence the correct answer will be (B). In the ajoining iagram if the capacity of each conenser is µf then the resultant capacity between the points P an Q will be - Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 8 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

19 Get Solution of These Packages & Learn by Vieo Tutorials on FREE Downloa Stuy Package from website: & Ex. 4 (A) (B) zero () µf (D) 8µF The ifferent branches of the conensers have number of conensers in G.P., hence the equivalent capacity µf. Hence the correct answer will be (). 7.5 omplex Mixe ombination : The given combination can not be simplifie by series or parallel combination such a combination is solve by using kirchhoff's law an other techniques. A special case of this combination is Wheatstone brige. 7.6 Wheat store brige : If (i) 4 5 or 5 4 then Such combination is calle balance Wheatstone brige. (ii) In this case V B V D. (iii) hange on (iv) (v) eq The equivalent can be converte in to given circuits: ( 4 )( 5 5 or ) Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 9 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

20 Get Solution of These Packages & Learn by Vieo Tutorials on FREE Downloa Stuy Package from website: & (vi) If 4 5 then eq 45 Ex.5 The equivalent capacity between the points X an Y in the following circuit (Figure) will be - Ex.6 (A) 6 mf (B) µf () 4 µf (D) µµf 4 5 Because the brige is balance, hence the central capacitance between Z an T is ineffective. an are connecte in series, hence their resultant µf similarly an 4 are connecte in series, hence their resultant µf. Now the two branches are connecte in parallel eq. 6µF Hence the correct answer will be (A). 7.7 Other important circuit solving techniques : (Applicable in both capacitive an resistive networks). (a) Equipotential Technique All the junctions which are at equal potential (such as junctions connecte by a connecting wire) can be replace by a single junction. So reraw the circuit to get it simplifie (b) Infinite ircuits Assume equivalent capacitance/resistance to be eq /R eq of whole network, then a one more branch (repetitive) in infinite network. alculate equivalent of this new circuit. It shoul again be equal to eq /R eq. Note:- If all the resistance/capacitances of a circuit are mae K times then equivalent will also become K-times. In the ajoining figure, the effective capacity of the group of conensers will be- (A) n (B) () zero (D).6 As the combination is spreaing upto infinity, hence the capacity will remain same at last but one step also. Let the capacity of the combination is. Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

21 Get Solution of These Packages & Learn by Vieo Tutorials on FREE Downloa Stuy Package from website: & [ ] or or This is a quaratic equation in ± Negative capacity is impossible.6 Hence the correct answer will be (D). () Kirchhoff's Laws (i) Junction Rule 5 Sum of charges present on the plates of capacitors connecte at a junction is equal to zero (If initially all the capacitors are uncharge) (while aing charges of ifferent plates battery can be neglecte as net charge on battery is zero). (ii) Loop Rule In any close loop the algebraic sum of potential rops across ifferent elements is equal to zero. 8. HARGING AND DISHARGING OF A AP APAIT ITOR 8. harging of a conenser : (i) (ii) (iii) (iv) (v) In the following circuit. If key is close then the conenser gets charge. Finite time is taken in the charging process. The quantity of charge at any instant of time t is given by q q [ e (t/r) ] Where q maximum final value of charge at t. Accoring to this equations the quantity of charge on the conenser increases exponentially with increase of time. If t R τ then q q [ e (R/R) ] q e or q q (.7).6 q 6% of q Time t R is known as time constant. i.e. the time constant is that time uring which the charge rises on the conenser plates to 6% of its maximum value. The potential ifference across the conenser plates at any instant of time is given by V V [ e (t/r) ] volt The potential curve is also similar to that of charge. During charging process an electric current flows Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

22 FREE Downloa Stuy Package from website: & Get Solution of These Packages & Learn by Vieo Tutorials on (vi) in the circuit for a small interval of time which is known as the transient current. The value of this current at any instant of time is given by Ι Ι [e (t/r) ] ampere Accoring to this equation the current falls in the circuit exponentially (Fig.). If t R τ Time constant Ι Ι Ι e ( R/R) e.7 Ι 7% of Ι i.e. time constant is that time uring which current in the circuit falls to 7% of its maximum value. Derivation of formulae for charging of capacitor it is given that initially capacitor is uncharge. let at any time Applying kirchoff voltage law ir q ir i q q R q t Rq q t Rq R. q t. q q q q O t t R O ln ( q) ln R t ln e q t/r q. e t/r q ( e t/r ) R time constant of the R series circuit. Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

23 Get Solution of These Packages & Learn by Vieo Tutorials on After one time constant FREE Downloa Stuy Package from website: & q e (.7).6. urrent at any time t q i t e R e t / R t /R R Voltage across capacitor after one time constant V.6 Q V V e( e t/r ) Voltage across the resistor V R ir e t/r Heat (by energy conservation work one by battery Ucapacitor ) c() ( ( ) c Alternatively : Heat H i Rt R R R R R t e R e t /R t e R /R R t t t e R In the figure time constant of () is more than (). Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

24 Get Solution of These Packages & Learn by Vieo Tutorials on Ex.7 Without using the formula of equivalent. Fin out charge on capacitor an current in all the branches as a function of time. FREE Downloa Stuy Package from website: & Applying KVL in ABDEA ir i R q q R q R q q q q q t R t R ( q) q q e t/r q ( e t/r ) t R q q ( e t/r ) i q q ( e t/r ) i Alternate solution by equivalent Time constant of circuit R R maximum charge on capacitor R R e t/r e t/r Hence equations of charge an current are as given below q ( e t/r ) q q ( e t/r ) i q q ( e t/r ) i R R e t/r e t/r Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 4 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

25 FREE Downloa Stuy Package from website: & Get Solution of These Packages & Learn by Vieo Tutorials on Ex.8 A capacitor is connecte to a V battery through a resistance of Ω. It is foun that the potential ifference across the capacitor rises to 4. V in µs. Fin the capacitances of the capacitor. The charge on the capacitor uring charging is given by Q Q ( e t/r ). Ex.9 Hence, the potential ifference across the capacitor is V Q/ Q / ( e t/r ). Here, at t µs, the potential ifference is 4V whereas the steay potential ifference is Q / V. So, 4V V( e t/r ) or, e t/r or, e t/r or, t R or, R Ιn.45 t.45 µs µ s or,.5 µf. Ω.469 µs Initially the capacitor is uncharge fin the charge on capacitor as a function of time, if switch is close at t. Applying KVL in loop ABDA ir (i i ) R ir i R Applying KVL in loop ABEFDA ir i R q i R i R q i R q q q. R t q t q q t R q ln q / R ( e ) t t R Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 5 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

26 FREE Downloa Stuy Package from website: & Get Solution of These Packages & Learn by Vieo Tutorials on Metho for objective : In any circuit when there is only one capacitor then t / τ q Q st ( e ) τ R eff. ; Q st steay state charge on capacitor (has been foun in article 6 in this sheet) Reffective is the resistance between the capacitor when battery is replace by its internal resistance. 8. Discharging of a conenser : (i) (ii) (iii) In the above circuit (in article 8.) if key is opene an key is close then the conenser gets ischarge. The quantity of charge on the conenser at any instant of time t is given by q q e (t/r) i.e. the charge falls exponentially. If t R τ time constant, then q e q.7q 7% of q i.e. the time constant is that time uring which the charge on conenser plates ischarge process falls to 7% (v) (vi) (iv) The imensions of R are those of time i.e. MºLºT an the imensions of R are those of frequency i.e. M L T. The potential ifference across the conenser plates at any instant of time t is given by V V e (t/r) Volt. The transient current at any instant of time is given by Ι Ι e (t/r) ampere. i.e. the current in the circuit ecreases exponentially but its irection is opposite to that of charging current. Derivation of equation of ischarging circuit : Applying K.V.L. q ir i R q q Q t q t q R Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 6 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

27 Get Solution of These Packages & Learn by Vieo Tutorials on ln Q q R t FREE Downloa Stuy Package from website: & Ex. q Q. i q t / R e t Q e R t / R At t S w is close, if initially is uncharge an is charge to a potential ifference then fin out following (Given ) (a) (b) (c) harge on an as a function of time. Fin out current in the circuit as a function of time. Also plot the graphs for the relations erive in part (b). Let q charge flow in time 't' from the battery as shown. The charge on various plates of the capacitor is as shown in the figure. Now applying KVL q q ir q q ir q ir q ir ir q q R q t q q q t ln( q) t R t R q t ln R q / R e t ( e t/r ) q q ( e t/r) Ans. Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 7 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

28 Get Solution of These Packages & Learn by Vieo Tutorials on q i e t/r t R Ans. FREE Downloa Stuy Package from website: & Ex. on the plate B q q e t/r e t/r / R [ e ] t The electric fiel between the plates of a parallel plate capacitance. µf rops to one thir of its initial value in 4.4 µs when the plates are connecte by a thin wire. Fin the resistance of the wire. The electric fiel between the plates is or, E Q A E E e t/r Q A e t/r In the given problem, E E at t 4.4 µs. Thus, or, e 4.4µ s R 4.4µ s In. R 4.4µ s or, R. Ω...µ F Miscellaneous Example : Ex. Two parallel conucting plates of a capacitor of capacitance containing charges Q an Q at a istance apart. Fin out potential ifference between the plates of capacitors. apacitance Electric fiel V Q A V Q Q A Q Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 8 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

29 FREE Downloa Stuy Package from website: & Get Solution of These Packages & Learn by Vieo Tutorials on 8. APAIT ITORS WITH DIELETRI (i) (ii) In absence of ielectric E When a ielectric fills the space between the plates then molecules having ipole moment align themselves in the irection of electric fiel. r b inuce charge ensity (calle boun charge because it is not ue to free electrons). * For polar molecules ipole moment * For non-polar molecules ipole moment For non-polar molecules the molecule of substance arrange as given below : (iii) σa V apacitance is presence of ielectric σ A σ. K AK AK Here capacitance is increase by a factor K. AK (iv) Polarisation of material : When nonpolar substance is place in electric fiel then ipole moment is inuce in the molecule. This inuction of ipole moment is calle polarisation of material. The inuce charge also prouce electric fiel. σ b inuce (boun) charge ensity. E in E E in σ σ b Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 9 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

30 FREE Downloa Stuy Package from website: & Get Solution of These Packages & Learn by Vieo Tutorials on It is seen the ratio of electric fiel between the plates in absence of ielectric an in presence of ielectric is constant for a material of ielectric. This ratio is calle 'Dielectric constant' of that material. It is represente by r or k. σ K σ b σ K (v) ase : () ase : () If the meium oes not fille between the plates completely then. Electric fiel will be as shown in figure The total electric fiel prouce by boun inuce charge on the ielectric outsie the slab is zero because they cancel each other. (vi) omparison of E (electric fiel), σ (surface charges ensity), Q (charge ), (capacitance) an F (force between the plates) before an after inserting a ielectric slab between the plates of a parallel plate capacitor. A Q V E V σ v A ' A K Q' 'V E' V also σ v K A Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

31 Get Solution of These Packages & Learn by Vieo Tutorials on Here potential ifference between the plates, Here potential ifference between the plates E V E FREE Downloa Stuy Package from website: & Ex. Note Ex.4 Equating both σ K σ' Kσ σ' E V V σ E V V σ' K In the presence of ielectric, i.e. in case IIn capacitance of capacitor is more. (vii) Energy ensity of a ielectric re If a ielectric slab of thickness t an area A is inserte in between the plates of a parallel plate capacitor of plate area A an istance between the plates ( > t) then fin out capacitance of system. What o you preict about the epenence of capacitance on location of slab? Q σa v v V V σ σ t σt K t t σ t k σ t t Q σa t k A t t /K ( t t t) (i) apacitance oes not epen upon the position of ielectric (it can be shifte up or own still capacitance oes not change). (ii) If the slab is of metal slab then A t Fin out capacitance between A an B if two ielectric slabs of ielectric constant K an K of thickness an an each of area A are inserte between the plates of parallel plate capacitor of plate area A as shown in figure. Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

32 Get Solution of These Packages & Learn by Vieo Tutorials on FREE Downloa Stuy Package from website: & Ex.5 σa V A K K ; V E E AK σ K AK σ K σ k k This formula suggests that the system between A an B can be consiere as series combination of two capacitors. Fin out capacitance between A an B if two ielectric slabs of ielectric constant K an K of area A an A an each of thickness are inserte between the plates of parallel plate capacitor of plate area A as shown in figure. A K, A K V σ V σ E K, E K σ K V σ σ σ K K σ b σ ( σ b σ ( K V ) k ) k The combination is equivalent to : Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

33 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't. Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page Get Solution of These Packages & Learn by Vieo Tutorials on FREE Downloa Stuy Package from website: & Ex.6 Fin out capacitance between A an B if three ielectric slabs of ielectric constant K of area A an thickness, K of area A an thickness an K of area A an thickness are inserte between the plates of parallel plate capacitor of plate area A as shown in figure. (Given istance between the two plates ) It is equivalent to K A K A K A K A. K A K A K A K A K K A K A K K K K A Ex.7 A ielectric of constant K is slippe between the plates of parallel plate conenser in half of the space as shown in the figure. If the capacity of air conenser is, then new capacitance between A an B will be- (A) (B) K () [ K] (D) K] [ This system is equivalent to two capacitors in parallel with area of each plate A. )K (A / / A A [ K] [ K] Hence the correct answer will be ().

34 Get Solution of These Packages & Learn by Vieo Tutorials on Ex.8 In the ajoining figure two ielectrics of constants K an K are fille in a parallel plate conenser. The capacity of the conenser will be : FREE Downloa Stuy Package from website: & Ex.9 K Ex.4 (A) () A (K K) K K A (B) KK (D) K K K K A (K K ) Two capacitors, each of area A an plate separation, are connecte in series. K A K A K K A K K A (K K ) The capacity of a parallel plate capacitor in air is 5µF an on immersing it into oil it becomes µf. The ielectric constant of oil is - (A).45 (B).55 (). (D). K. 5 5 Two parallel plate conensers with capacities an are connecte in parallel an are charge to potential ifference V. Now the battery is remove an a ielectric of constant K is inserte between the plates of conenser. Now the potential ifference across each conenser will be- V (A) K q V V (B) When ielectric is inserte K (K ) V K the potential ifference across the capacitors V f total charge total capacitance q V " ( K) V K () V K (D) K V Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 4 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

35 FREE Downloa Stuy Package from website: & Get Solution of These Packages & Learn by Vieo Tutorials on Ex.4 Ex.4 A parallel plate conenser is charge to a certain potential an then isconnecte. The separation of the plates is now increase by.4 mm an a plate of thickness mm is inserte into it keeping its potential constant. The ielectric constant of the meium will be- (A) 5 (B) 4 () (D) As charge an potential of the conenser both are constant in two cases, hence its capacity must also remain constant or A A ' t K or t K or ( ) t K or.4 K or K.8 or K. K 5 Three ielectric materials are fille in a parallel plate conenser as shown in figure. A potential ifference is applie between the plates of the conenser. If the area of each plate is S an istance between parallel plates is then the capacity between the point X an Y will be - (A) S K K K K K K S / / K S / / S K an are in series an are in parallel K S KK ( S) / K S KS (B) S () SKK K (D) SK K Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 5 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

36 Get Solution of These Packages & Learn by Vieo Tutorials on FREE Downloa Stuy Package from website: & S K K K K K Hence the correct answer will be (A). (viii) Force on a ielectric ue to charge capacitor If ielectric is completely insie the capacitor then force is equal to zero. ase I - Voltage source remains connecte where F V constant. U V U F x F xb K b b (K ) V x ( l x)b [Kx l x] b(k )V constant (oes not epen on x) Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 6 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

37 Get Solution of These Packages & Learn by Vieo Tutorials on ase II : When charge on capacitor is constant FREE Downloa Stuy Package from website: & Ex.4 Ex.44 xb K U Q F x ( l x)b, U Q. x. x Q x b (K l) Fin V an E at : ( Q is a point charge kept at the centre of the nonconucting neutral thick sphere of inner raius 'a' an outer raius 'b') (i) < r < a (ii) a r < b (iii) r b q an q charge will inuce on inner an outer surface respectively KQ E( < r < a) r E (r b) E (a r < b) V (r b) r KQ r KQ Kq r KQ r r q Q. r KQ r r KQ kq kq (a r b) V A V P ( r) r b r b KQ V (r a)v B V ( r) r r a b r kq b r r kq kq a b r a What is potential at a istance r (<R) in a ielectric sphere of uniform charge ensity ρ, raius R an ielectric constant r.. V A V B W B A q Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 7 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

38 Get Solution of These Packages & Learn by Vieo Tutorials on FREE Downloa Stuy Package from website: & Q ρr V 4π R ( r) r Q 4π R V outisie KQ r r R r ρ(r r ) 9. OMBINATION OF PARALLEL PLATES Ex.45 Fin out equivalent capacitance between A an B. Put numbers on the plates The charges will be as shown in the figure. V V V 4 so all the capacitors are in parallel combination. eq Ex.46 Fin out equivalent capacitance between A an B. These are only two capacitors. eq Ex.47 Fin out equivalent capacitance between A an B. The moifie circuit is Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 8 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

39 Get Solution of These Packages & Learn by Vieo Tutorials on FREE Downloa Stuy Package from website: & Other metho : eq V Q xa V eq V V V 4 (V V ) (V V ) x x x xax eq x A. Ex.48 Fin out equivalent capacitance between A an B. eq eq A 5 Alternative Metho : V Q 5 A 5 x y VAB Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 9 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

40 Get Solution of These Packages & Learn by Vieo Tutorials on FREE Downloa Stuy Package from website: & Ex.49 V Q x y VAB Potential ifference between an 4 is same y A y x V x A y x A ( x x)xa (5x) A. 5 Five similar conenser plates, each of area A, are place at equal istance apart an are connecte to a source of e.m.f. E as shown in the following iagram. The charge on the plates an 4 will be- (A) () A, AV, A AV Equivalent circuit iagram harge on first plate Q V Q AV harge on fourth plate Q ( V) Q AV (B) (D) A, AV, AV As plate 4 is repeate twice, hence charge on 4 will be Q Q Q AV Hence the correct answer will be (B). 4 AV Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 4 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.

41 FREE Downloa Stuy Package from website: & Get Solution of These Packages & Learn by Vieo Tutorials on OTHER TYPES OF APAIT ITORS Spherical capacitor : This arrangement is known as spherical capacitor. KQ KQ KQ KQ V V a b b b If b >> a Q V V 4π ab b a 4π a KQ KQ a b Q KQ KQ a b ac K(b a) 4πr ab b a ylinrical capacitor 4π ab b a There are two co-axial conucting cylinrical surfaces l >> a >> b where a an b is raius of cyliners. V λ λ a Kλln b 4π b ln a π b ln a apacitance per unit length π b ln a F/m Teko lasses, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 4 Successful People Replace the wors like; "wish", "try" & "shoul" with "I Will". Ineffective People on't.