Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,

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1 PCE CURRENT ELECTRICITY C Electric Current Current is rate of flow of charge from one region to another. Mathematically, dq I where dq is net dt q charge transported at section during time dt. For steady state, I. The SI unit of current is t ampere (A). When A current flows through a conductor, then electrons flow per second.. If the current will constitute of -particles then the number of -particles flowing per second for µa current is If the current is given by at + b where a and b are constant, then the charge flown in time t is at bt at bt at + bt at bt 3. If the charge flowing is given by exp ( at) then the current is ae at ae at e at none [Answers : () () (3) ] CA CB Current in Different Materials Current in conductors is due to motion of electrons, in electrolytes due to motion of both positive and negative ions, in semi-conductors due to motion of electrons and holes. Electric Current in Conductors There is a net current in the conductor is given by I = nev d A where n is number of electrons per unit volume, A is the cross-section area of conductor, e is the electric charge and v d is its drift speed. The drift velocity is very small (~ 4 m/s) as compared to thermal speed of electrons at room temperature (~ 5 m/s). The current per unit cross section is called the current density j, given by whereas I is a scalar. j nev.note that j is a vector. The average drift speed of conduction electrons in a copper wire of cross-sectional area. 7 m carrying a current of.5 A (Assume that each copper atom contributes roughly one conduction electron. The density of copper is 9. 3 kg/m 3, and its atomic mass is 63.5 u).. mm/s. mm/s 3.3 mm/s 4.4 mm/s [Answers : () ] d C3A Ohm s Law and Electrical Resistance I V V I where R is the electrical resistance of the conductor. Resistance For a linear homogenous R l conductor of uniform cross section A and of the length l the resistance is given by R where is A specific resistance or resistivity of the material. Resistivity depends on the material of the conductor and its temperature. Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

2 PCE. If the percentage change in the length of wire is % on stretching then the percentage change in resistance is % 3 % 4 % 5 %. If the percentage change in the length of wire is % on stretching then the percentage change in resistance is % % % 3 % 3. If the percentage change in the cross-sectional radius of wire is % on stretching then the percentage change in resistance is 4 % 6 % 8 % 8 % 4. A metallic wire of length m is stretched to double its length. The ratio of its initial and final resistance assuming that there is no change in its density on stretching is : : 4 : 4 : 5. The voltage-current graphs for two resistors of the same material and same radii with lengths L and L are shown in figure. If L > L, which of these graphs represent V I for L? graph A graph B both can be none [Answers : () c () b (3) c (4) b (5) b] C3B Resistance of special configuration : Configuration Resistance. Hollow cylinder Across it ends L (b a R ) Across its inner and outer surface R L b ln a. A Truncated Cone L R ab Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

3 PCE 3 3. A sphere (b a) R 4ab. The resistivity of a cylinder, of length l and radius r, changes with distance according to ax + b where a and b are constant quantity and x is the distance measured from left end. The resistance of cylinder if the potential drop across its ends al bl r al bl r. Find the resistance for the following configurations : [Answers : () a] al bl r none Cylinderical shell of inner radius a and outer radius b with length l consistsof a material of resistivity if the potential drop is between inner and outer shell. Cylinderical shell of inner radius a and outer radius b with length l consistsof a material of resistivity if the potential drop is across the ends. Spherical shell of inner radius a and outer radius b consists of a material of resistivity if the potential drop is between inner and outer shell. A truncated cone. C3C Another form of ohm s Law J E E where is the conductivity, reciprocal of resistivity and E is the electric field. C3D v = ne d neµ E, µ is known as mobility of the charge carrier. ne The conductivity is given by where n is the number of electrons per unit volume, e is the m electronic charge, is the relaxation time or collision time and m is the mass of electron. Dependence of resistivity and resistance on temperature T = [ + (T T )] where is the resistivity at reference temperature T and T is the resistivity at temperature T. The factor is called the temperature coefficient of resistivity. In the same way R T = R [ + (T T )]. V I graph for a given metallic wire at two temperatures are shown, which of these is for a higher temperature? Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

4 graph () graph () any graph none PCE 4. An electric toaster uses nichrome for its heating element. When a negligibly small current passes through it, its resistance at room temperature (7. C) is found to be When the toaster is connected to a 3 V supply, the current settles, after a few seconds, to a steady value of.68 A. The temperature coefficient of resistance of nichrome averaged over the temperature range involved, is.7 4 C. The steady temperature of the nichrome element is 547 C 647 C 747 C 847 C [Answers : () b () d] C4 C5 Resistivity of Various Materials The conductors has the resistivity in the order of 8 m, for carbon (graphite) has the resistivity m and has negative temperature coefficient for resistivity and for insulator the resistivity has the order greater than 5 m. There are two types of resistors : wire bound resistors and carbon resistors. Wire bound resistors are made by winding the wires of an alloy, viz, manganin, constantan, nichrome or similar ones. The choice of these materials is dictated mostly by the fact that their resistivities are relatively insensitive to temperature. These resistances are typically in the range of a fraction of an ohm to a few hundren ohms. Resistors in the higher range are made mostly from carbon. Carbon resistors are compact, inexpensive and thus find extensive use in electronic circuits. Carbon resistors are small in size and their values are given using a colour code. Analysing the Electrical Circuit Kirchoff s rules provides a general method for analyzing circuit networks. Law I Kirchoff s junction rule of kirchoff s current law (KCL) : KCL is based on conservation of charge principle. This algebraic sum of the current at any junction is zero i.e. I. Law II Kirchoff s loop rule of kirchoff s voltage Law (KVL) : This law is based on conservation of energy principle. The algebraic sum of the potential difference in any closed loop across each elements of the circuit must equal zero. That is V. Problem solving strategy for electric curciut using Kirchoff s Laws :. Assume currents with their direction in each branch of the circuits keeping in mind the KCL. Often you will not known in advance the actual direction of an unknown current but this doesn t matter. Carry out your solution, using the assumed direction. If the actual direction of a particulay quantity is opposite to your assumption, the result will come out with a negative sign.. Choose any closed loop in the network, and designate a direction (clockwise or countercolckwise) to travel around the loop in applying the KVL. Travel, adding potential difference as you cross them and equate them to zero. Remember the following rules : An emf is counted as positive when you transverse it from to + and negative when you transverse it from + to i.e., If we transverse through a battery from the negative terminal to the positive terminal, the change in voltage is +E. If we transverse through a battery from the positive terminal to the negative terminal, the change in voltage is E. An IR term is negative if you travel through the resistor in the same direction as the assumed current and positive if you travel in the opposite direction, i.e., Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

5 PCE 5 If a resistor is transverse in the same direction as the assumed current flow, the change in potentials is IR. If a resistor is transversed in the direction opposite from the assumed current flow, the change in potential is +IR. Remember that the potential difference across the terminal of a battery of emf E and internal resistance r is given by (i) when the current i is drawn (discharging battery), V = E ir (ii) when the current i is supplied (charging battery), V = E + ir.. The storage battery of a car has an emf of V. If the internal resistance of the battery is.4, the maximum current that can be drawn from the battery is 5 A 8 A 3 A 3 A. A battery of emf V and internal resistance 3 is connected to a resistor. If the current in the circuit is.5 A. The terminal voltage of the battery when the circuit is closed is 5.5 V 6.5 V 7.5 V 8.5 V [Answers : () c () d] C6 Combination of Resistances and Equivalent Resistance Series combination (Potential divider) : In general for any number of resistors in series, R eq R i Parallel Combination (Current divider) : For any combination of resistors in parallel,. A wire of resistance 4 R is cut into 4 identical parts. Each part is stretched to twice the original length. If all the four wires after stretching are connected in parallel. The new resistance of the combination is R R 3R 4R. In the circuit given below with E = 65 V, R = 5, R =, R 3 = and R 4 = 3. R eq n I n i R i Rhe voltage drop across the resistor R is V 5 V V 5 V Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

6 PCE 6 3. Consider the following circuit when the switch S is closed. The current passing through the connecting wire if the battery of V is connected across A and B A A 3 A none 4. A set of n identical resistors, each of resistance R, when connected in series have an effective resistance X and when the resistors are connected in parallel, their effective resistance is Y. The relation between R, X and Y is R XY R XY 3 R XY 4 R XY [Answers : () a () d (3) d (4) a] C7 Combination of Cells Series combination Here E = E + E + E and r = r + r + r Parallel combination Here E E E E r r r r r r 3 and 3 3 r r r r3 Mixed Combination Let mn identical cells are connected as shown in figure. Here E = emf of each cell r = internal resistance of each cell The combination of cells is equivalent to a single cell of emf = me and internal resistance For the maximum current the external resistance should be mr R is connected n. Two identical cells of emf.5 V each joined in parallel provide supply to an external circuit consisting of two resistors of 7 each joined in parallel. A very high resistance voltmeter reads the terminal voltage of the cells to be.4 V. The internal resistance of each cell is mr n Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

7 PCE 7. Three cells of emf. V,.8 V and.5 V are connected in series. Their internal resistances are.5,.7 and., respectively. If this battery is connected to an external resistor of 4. The potential difference across the terminals of the cell of emf.5 V while in use is.8 V.38 V.48 V.58 V 3. Two cells of emf.5 V and V and internal resistance and respectively are connected in parallel to pass current in the same direction through an external resistance of 5. The potential difference across the 5 resistor. 5 V 7 6 V 7 7 V 7 8 V 7 [Answers : () a () d (3) a] C8 Electrical Instruments. Galvanometer Galvanometer detects the presence of current in the branch where it is connected. It does not measure current.. Ammeter It is always connected in series with the branch in which it has to measure the current. Its resistance should be small enough so that its presence does not significally alter the current in the branch. The resistance of an ideal ammeter is zero. A galvanometer may be converted into an ammeter by connecting a low resistance (called shunt) in parallel to the galvanometer S. Ig The magnitude of shunt S resistance is given by S G I I where 3. Voltmeter G = resistance of the galvanometer I g = full scale deflection current of the galvanometer I = maximum current to the measured by the ammeter It is always connected in parallel with the branch across which it has to measure the voltage. Its resistance should be large enough so that it may draw very small current from the circuit; and the disturbance produces is significant. An ideal voltmeter has infinite resistance. A galvanometer may be converted into a voltmeter by connecting a large resistance in series with the galvanometer. The magnitude of the high resistance is given by voltage to be measure by the voltmeter. 4. Potentiometer Uses of Potentiometer Comparison of emf s g R V G I g, where V is the maximum The emf of an unknown battery E may be obtained by comparing its length l for zero deflection to that of the unknown battery E with balance point located at l. Thus, E E l. l Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

8 Measurement of Internal Resistance PCE 8 The balance point l is obtained with only emf E connected. A second balance point l is obtained by l connection of external resistance R. Then internal resistance is given by r R l. In a potentiometer arrangement, a cell of emf.5 V gives a balance point at 35. cm length of the wire. If the cell is replaced by another cell and the balance point shifts to 63. cm, the emf of the second cell is.5 V.5 V.5 V.5 V. A. V potentiometer used for the determination of internal resistance of a.5 V cell. The balance point of the cell in open circuit is 76.3 cm. When a resistor of 9.5 is used in the external circuit of the cell, the balance point shifts to 64.8 cm length of the potentiometer wire. The internal resistance of the cell is A m long wire of uniform cross-sectional and resistance is used in a potentiometer. The wire is connected in series with a battery of 5 V along with an external resistance of 48. If an unknown emf E is balanced at 6. m length of the wire. The potential gradient of the potentiometer wire is V/m 3 V/m 4 V/m 5 V/m 4. In the above problem, the value of unknown emf is. V. V.4 V.6 V 5. The length of the potentiometer wire is 6 cm and it carries a current of 4 ma. For a cell of emf V and internal resistance, the null point is found to be at 5 cm. If a voltmeter is connected across the cell, the balancing length is decreased by cm. The resistance of the voltmeter is Four identical cells, each of emf V, are joined in parallel providing supply of current to external circuit consisting of two 5 resistors joined in parallel. The terminal voltage of the cell as read by an ideal voltmeter is.6 V. The internal resistance of each cell is 3 none 7. A voltmeter V of resistance 4 is used to measure the potential difference across a resistor in the circuit in which an emf of 84 V is connected with resistor in series. The reading on the voltmeter is V V 4 V 6 V [Answers : () c () b (3) a (4) b (5) b (6) d (7) c] C9 C Wheatstone Bridge Let us assume that R is unknown and R 3 can be varies and it is adjusted until no current flows in the galvanometer. In this condition, wheatstone bridge is said to be balanced, and the resistances satisfy the ratio R R R R Meter Bridge 3 4 It is used to measure the unknown value of a resistor which is based on Wheatstone bridge. The four arms AB, BC, DA and CD [with resistances R, S, l and ( l)] form a Wheatstone bridge where is known as resistance per unit length of the wire. If the jockey is moved along the wire, then there will be one position where the galvanometer will show no current. Let the distance of the jockey from the l end A at the balance point be l = l. The balance condition gives R S. l Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

9 PCE 9. In a meter bridge, the null point is found at a distance of 33.7 cm. If now a resistance of is connected in parallel with S, the null point occurs at 5.9 cm. The value of R is none. In the circuit, a metre bridge is shown in its balanced state. The metre bridge wire has a resistance of ohm/cm. The value of the unknown resistance is For the balance wheat stone bridge, the value of X is [Answers : () a () b (3) b] C Energy and Power In Electrical Circuits Rate of delivering electrical energy or power input to a circuit element, P = V ab I Power delivered to a resistor = VI = I R = V R Power output of a source = V ab I = E I I r Power input to a source : P = V ab I = (EI + I r) Maximum Power Transfer Theorem The maximum power is transferred to an external resistor if its resistance is equal to the internal resistance of the cell of battery.. Which one has the greatest resistance? V, 6 W bulb V, W bulb V, kw heater all have same resistances. Two bulbs whose resistances are in the ratio of : are connected in parallel to a source of constant voltage. The ratio of power dissipation in these bulbs is : : : 4 4 : Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

10 PCE 3. Two heater wires of the same dimensions are first connected in series and then in parallel to a source of supply. The ratio of heat produced in the two cases is : : : 4 4 : 4. An electric motor operating on a 5 V d.c. supply draws a current of A. If the efficiency of the motor is 3%, the resistance of the windings of the motor is none 5. Three identical resistors, each of resistance R, when connected in series a d.c. source, dissipate power X. If the resistors are connected in parallel to the same d.c. source, then the power will be dissipated is 9X 6X 3X X 6. A heater coil is rated W, V. It is cut into two identical parts. Both parts are connected together in parallel to the same source of V. The energy liberated per second in the new combination is W W 3 W 4 W 7. Four cells of identical emf E, internal resistance r, are connected in series to a variable resistor. The graph shows the variation of terminal voltage of the combination with the current output : The current from the cells for which maximum power dissipation occur in the circuit is.5 A.75 A A.5 A [Answers : () a () b (3) c (4) a (5) a (6) d (7) c] C Capacitance and type of Capacitors : The capacitance C of a capacitor is defined from q = CV, where V is the potential difference the conductors. The SI unit of capacitance is the farad, ( farad = F = CV ). (i) Parallel Plate Capacitors without dielectric : C = (ii) Parallel Plate Capacitors with dielectric : C = d A d K A where K is dielectric constant of dielectric within the capacitor, A is the area of the plate and d is the distance between the plates. (iii) Spherical Capacitor : C 4 b ab a If the radius of the outer sphere tends to infinity b, the capacitance reduces to C = 4 a which is called the capacitance of an isolated sphere. Here a and b are the radius of inner and outer sphere. (iv) Cylindrical Capacitor : C = cylinder and l is length of the cylinder. l b ln a where a and b are inner and outer radius of the Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

11 . The ratio of electric field at any two points in the parallel plate capacitor is : : : : 3 PCE. The given graph shows the variation of charge q versus potential difference V for two capacitors C and C. The two capacitors have same plate sepration but the plate area of C is double than that of C. Which of the lines in the graph correspond to C? A B both may be none [Answers : () a () b] C3 Batteries and Capacitors A battery maintains a certain potential difference its terminals. Thus, when capacitors is connected to a battery, charge flows between the capacitor and the battery until the capacitor has the same potential across it as the battery. When a charged capacitor disconnected from the battery and then connected to a uncharged capacitor, the potential difference across the charged capacitor is changed. But the total charge of the capacitors system is constant.. You are given an isolated parallel plate capacitor of capacitance C charged to a potential difference V. Which of the following quantity remains same when the separation between the plates is doubled with the help of insulating handles attahced to the plates? potential difference across the plates field between the plates energy stored in the capacitor none [Answers : () d] C4 Combination of Capacitors Capacitors in Parallel When a potential difference V is applied across several capacitors connected in parallel, the potential difference V is applied across each capacitor. The total charge q stored on the capacitor is the sum of the charges stored on all the capacitors. Capacitors connected in parallel can be replaced with an equivalent capacitor that has the same total charge q and the same potential difference V as the actual capacitors. C eq for n capacitor in parallel is given by n C eq C j j Capacitors in Series When a potential difference V is applied across several capacitors connected in series, the capacitors have identical charge q. The sum of potential differences across all the capacitors is equal to the applied potential difference V. Capacitor that connected in series can be replaced with an equivalent capacitor that has the same charge q and the same potential difference V as the actual series capacitors. C eq for n capacitors in series is given by C eq n j C j Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

12 PCE. Three capacitors of equal capacitance, when connected in series, have a net capacitance of C and when connected in parallel, have a capacitance of C. The value of C /C is : 9 9 : : 3 3 :. When two capacitors of capacitance C and C are connected in series the net capacitance is 3 µf; when connected in parallel its value is 6 µf. The value of C is µf µf 3 µf none 3. The equivalent capacitance of the following network between A and B is.5 µf 5 µf.66 µf none 4. The equivalent capacitance of the following network between A and B is.5 µf 5 µf.66 µf none 5. The equivalent capacitance of the following network between A and B is.5 µf 5 µf.66 µf none 6. An electrical technician requires a capacitance of µf in a circuit across a potential difference of kv. A large number of µf capacitors are available to him each of which can withstand a potential difference of not more than 4 V. Suggest a possible arrangement that requires the minimum number of capacitors. The minimum number of capacitors is Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

13 PCE 3 7. The potential difference across C 4 is V V 5 V none 8. A spherical capacitor has an inner sphere of radius cm and an outer sphere of radius 3 cm. The outer sphere is earthed and the inner sphere is given a charge of.5 µc. The space between the concentric spheres is filled with a liquid of dielectric constant 3. The capacitance of the capacitor is.5 nf 3.5 nf 4.5 nf 5.5 nf 9. In the above problem the potential of the inner sphere is 5 mv 35 mv 45 mv 55 mv. The ratio of the capacitance of this capacitor with that of an isolated sphere of radius cm is A cylindrical capacitor has two co-axial cylinders of length 5 cm and radii.5 cm and.4 cm. The outer cylinder is earthed and the inner cylinder is given a charge of 3.5 µc. The potential of the inner cylinder is.9 4 V V V V. A parallel plate capacitor is to be designed with a voltage rating kv, using a material of dielectric constant 3 and dielectric strength about 7 V m. (Dielectric strength is the maximum electric field a material can tolerate without breakdown i.e., without starting to conduct electricity through partial ionisation). For safety, we should like the field never to exceed, say % of the dielectric strength. The minimum area of the plates is required to have a capacitance of 5 pf is 7 cm 8 cm 9 cm cm [Answers : () a () d (3) a (4) b (5) c (6) b (7) b (8) d (9) c () d () a () c] C5 Energy Stored in Capacitor The energy stored in a capacitor is given by U CV Q C. The total energy stored in the capacitors in the given network is.6 5 J.6 5 J J J Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

14 PCE 4. X and Y are two parallel plate capacitors having the same area of plates and same separation between the plates. X has air between the plates and Ycontains a dielectric medium of r = 5. The ratio of electrostatic energy stored in X and Y is : 3 : 4 : 5 : [Answers : () c () d] C6 Capacitor with a Dielectric When the space between the conductors is filled with a dielectric material, the capacitance increased by a factor K, called the dielectric constant of the material i.e. C = K C Capacitor with dielectric insertion when battery is disconnected (i) Charge remains constant i.e. q = q (ii) (iii) Potential between the plates decreases i.e. Field between the plates decreases i.e. V V K E E K (iv) Energy stored in the capacitor decreases i.e. U U K Capacitor with dielectric insertion when battery is connected. (i) Potential difference remains constant. (ii) Charge on plates increases i.e. q = Kq (iii) Electric field remains constant (iv) Energy stored in the capacitor increases i.e. U = KU. In a parallel plate capacitor, the capacitance increase from 4 microfarad to 8 microfarad, on introducing a dielectric medium between the plates. The dielectric constant of the medium is 3 4 [Answers : () b] C7 Dielectrics and polarization If the medium between the plates of a capacitor is filled with an dielectric (insulating substance), the electric field due to charged plates induces a net dipole moment in the dielectric. This effect is called polarisation. Although the net charge in the dielectric is zero but an electric field rises in the dielectric due to polarisation which is opposite to the external field E. The electric field in the dielectric is E/K where K is the dielectric constant of the dielectric. Hence the potential difference between the plates is reduced and that s why the capacitance will increase due to introduction of the dielectric by the factor K. The magnitude of induced charge on dielectric is given by q q P on dielectric and q is the charge on the plate of the capacitor. K, where q P is the induced charge Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

15 PCE 5 C8 Force Between the Plates of Capacitor The plates of parallel plate capacitor attracts each other with a force given F q. A Electrostatic stress of force per unit area acting on either capacitor plate is given by E.. A conducting plate A of surface area. m is suspended from one arm of a sensitive balance such that it is parallel to another similar horizontal plate B and is at a height of mm above B. The plate A in this situation is balanced by some weight in the other arm of the balance. A potential difference of V is established between plates A and B. The additional mass should be added to the other arm to keep the system balanced is.3 gm.3 gm.33 gm.44 gm. A parallel plate capacitor of plate area A and charge q is connected to a spring of spring constant k. The other end of the spring is connected to the wall. The extension in the spring at equilibrium is q AK q 3 AK q 4 AK q 5 AK [Answers : () a () a] C9 (i) (ii) Force on Dielectrics A charged parallel plate capacitor, with battery disconnected or battery connected, pulled the dielectric inwards. The force acting on a dielectric slab while battery connected is given by F V dc dx The force acting on the dielectric slab while battery disconnected is given by q C F dc dx. A parallel plate capacitor with plates width b and length L, the separation between the plates d is held constant and the plates are connected to a battery of potential difference V. A dielectric slab of relative permittivity K and thickness equal to the separation between the plates is inserted. Calculate the electrostatic energy of the system and force on the slab when x length of the slab is already inserted into the space between the plates. bv bv [Answers : () [L x(k )] (K ) ] d d Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

16 C Capacitors with Electric Circuit Applying Kirchoff s rule for circuits with capacitors : PCE 6. If we traverse through a capacitor from the negative terminal to the positive terminal, the change in potential is q. C. If we traverse through a capacitor from the positive terminal to the negative terminal, the change in potential is q. C. In the circuit shown find the charge on each capacitor. [Answers : () 4µC, 8µC, 6µC] C Circuits with Resistors and Capacitors : Charging of a Capacitor The figure shows a circuit arrangement in which a capacitor in series with a resistor is connected across a battery of emf E through a switch S. When the switch S is closed the current starts flowing and the capacitor starts charging. If i be the instantaneous current in the circuit and q be the instantaneous charge on capacitor, then applying Kirchoff s q C Voltage Law, we get ir E After solving the equation we get the following results. q = Q max ( e t/ ), v = V max ( e t/ ), i = I max e t/ Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

17 PCE 7 where Q max = CE; V max = E; I max = E/R and = RC (time constant) Charge on a capacitor is increasing with time q = CE( e t/rc ). At t =, q is zero and as t =, q is Q max = CE Potential difference across the capacitor increases with time v = E( e t/rc ). At t =, v is zero and at t =, v is V max = F Current through the capacitor decreases with time E i e R t/rc. At t =, i = I max = E/R and t =, i = The time constant is the time during which charge on the capacitor rises to.63 times the maximum value Mathematically, when t =, q = q max ( e ) or q =.63 q max. 37 e. For the charging RC-circuit draw the graph between lni vs. time t and what is the physical meaning of slope of this graph.. A 5. k resistor and a capacitor are connected in series and then a. V potential difference is suddenly applied across them. The potential difference across the capacitor rises to 5. V in.3 µs. Calculate the time constant of the circuit. Find the capacitance of the capacitor. 3. A 3. M resistor and a. µf capacitor are connected in series with an ideal battery of emf = 4. V. At. s after the connection is made, what are the rates at which the charge of the capacitor is increasing, energy is being stored in the capacitor, thermal energy is appearing in the resistor, and energy is being delivered by the battery? 4. An initially uncharged capacitor C is fully charged by a device of constant emf connected in series with a resistor R. Show that the final energy stored in the capacitor is half the energy supplied by the emf device. By direct integration of i R over the charging time, show that the thermal energy dissipated by the resistor is also half the energy supplied by the emf device. 5. In the following circuit initially the capacitor is uncharged. Find the current passing through the battery at any time t. Also find the current through the battery at t = and t =. [Answers : ().4 µs 6 pf (3).955 µc/s.8 µw.74 µw 3.8 µw] Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

18 Discharging of a Capacitor : PCE 8 The figure shows a capacitor C with an initial charge Q connected in series with a resistor. When the switch is closed the capacitor starts discharging. If i be the instantaneous current and q be the instantaneous charge on the capacitor then, using Kirchoff s voltage law q C dq R dt or dq q dt RC After solving the above equation, we get q = Q e t/, i = I e t/, V = V e t/ where = RC is the time constant, Q, I and V are the initial charge, current and potential difference across the capacitor.. A. µf capacitor with an initial stored energy of.5 J is discharged through a. M resistor. What is the initial charge on the capacitor? What is the current through the resistor when the discharge starts? Determine V C, the potential difference across the capacitors, and V R, the potential difference across the resistor, as functions of time. Express the production rate of thermal energy in the resistor as a function of time.. Consider a discharging RC circuit. Draw the lnq vs. time graph and what is the physical meaning of the slope. 3. How many time constants will elapse before the charge on a capacitor falls to.% of its maximum value in a discharging RC circuit? [Answers : (). 3 C 3 A V C = 3 e t V, V R = 3 e t V; P = e t W] Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

19 PCE 9 SINGLE CORRECT CHOICE TYPE. Masses of three wires of copper are in the ratio : 3 : 5 and their lengths are in the ratio 5 : 3 :. The ratio of their electrical resistances is : 3 : 5 5 : 3 : : 5 : 5 5 : 5 :. An infinite ladder network is constructed with and resistors as shown. The effective resistance between A and B is 7. A potentiometer wire of length cm has a resistance of. It is connected in series with a resistance and an accumulator of emf V and of negligible internal resistance. A source of emf mv is balanced against a 4 cm length of the potentiometer wire. The value of the external resistance is In the circuit shown, the cell is ideal with emf = 5V, and each resistance is 6. The maximum potential difference across the capacitor is In the above problem, the current through the resistor nearest to the battery is.5 A. A 3. A 4. A 4. 4 identical cells, each of internal resistance.5, are arranged in a parallel combination of n rows, each row containing m cells in series. The combination is connected across a resistor of 3. In order to send maximum current through the resistor, we should have m =, n = m = 8, n = 3 m =, n = m = 3, n = 8 5. A and B are two points on a uniform ring of resistance R. The ACB =, where C is the centre of the ring. The variation of equivalent resistance between A and B with is constant linear parabolic exponential 6. Seven resistors, each of, are connected as shown in the figure. The effective resistance between A and B is 5 V V 9 V zero 9. In the network shown, each resistance is equivalent to R. The equivalent resistance between points A and B is R/3 R/3 R 4R/3. The time constant of the circuit shown in figure is Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

20 PCE 3 R C RC RC Not defined. A conductor with rectangular cross section has dimensions (a a 4a) as shown in figure. Resistance across AB is x, across CD is y and across EF is z. Then 3.7 V 6.7 V 9.8 V.5 V 5. A cylinder of radius R made of electrical conductivity is surrounded by a cylindrical shell of inner radius R and outer radius R made of a material of electrical conductivity. The two ends of the combined system are maintained at two different potentials. The equivalent electrical conductivity of the system is 3 4 x = y = z x > y > z y > z > x x > z > y. n identical cells are joined in series with two cells A and B with reversed polarities. EMF of each cell is E and internal resistance is r. Potential difference across cell A or B is (n > 4) In the circuit shown in figure, the battery is an ideal one with emf V. The capacitor has capacitance C is initially uncharged. The switch S is closed at time t =. Let at t = the current through the battery is I and after large time the current through the battery is I. E n E n 4E n E n 3. A metallic sphere of radius a is surrounded by a concentric thin metallic shell of radius b. The space between these two electrodes is filled with a homogenous poorly conducting medium of resistivity. An emf of voltage V is connected between the metallic shells. The current through the gap between the two electrodes is Then I /I equals to : 4 : 3 : 3 3 : 5 7. In the above problem the maximum energy on the capacitor is 4abV (b a) V(b a ) (b a) CV CV 4 V(b a ) (b a) V(b a ) (b a) 4. The potential difference between X and Y i.e. V XY in the circuit shown in figure is Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857 CV 8 CV 6 8. A parallel plate capacitor of value.77 µf is to be designed using a dielectric material (dielectric constant =, breakdown strength = 3 6 Vm ). In order to make such a capacitor, which can withsdand a potential difference of V across the plates, the separation between the plates d and the area A of the plates can be ( = 8.85 C N m ) d = 6 m, A = 3 m d = 5 m, A = m d = 4 m, A = 4 m d = 4 m, A = 5 m

21 PCE 9. The capacitance of a parallel plate capacitor is C. Now it is half filled with a dielectric of constant K as shown in figure, the new capacitance is C. Now the same capacitance is half filled with same dielectric as shown in figure. Then C : C : C 3 equals to K K : : K K K : : 3 K K K : : 4 K K K : : 5 K. Two parallel plate capacitors of capacitances C and C are connected in parallel and charged to a potential difference V. The battery is then disconnected and the region between the plates of the capacitor C is completely filled with a material of dielectric constant K. The potential difference across the capacitors now becomes 3V/(K + ) V/(K + ) 4V/(K + ) 6V/(K + ). Five identical capacitor plates, each of area A, are arranged such that adjacent plates are at a distance d apart. The plates are connected to a source of emf V as shown in the figure. The charge on plate 4 is AV d 3 AV d AV d 4 AV d. A capacitor of capacitance µf is charged to a potential 5 V with a battery. The battery is now disconnected and an additional charge µc is given to the positive plate of the capacitor. The potential difference across the capacitor will be 5 V 8 V V 6 V 3. A dielectric slab of thickness d is inserted in parallel plate capacitor whose negative plate is at x = and positive plate is at x = 3d. The slab is equidistant from the plates. The capacitor is given some charge. As x goes from to 3d, the magnitude of the electric field remains the same the direction of the electric field remains the same the electric potential increases continuously both and are correct 4. Two identical metal plates are given positive charges Q and Q ( < Q ) respectively. If they are now brought close together to form a parallel plate capacitor with capacitance C, the potential difference between them is : (Q + Q ) / (C) (Q + Q ) / C (Q Q ) / C (Q Q ) / (C) 5. A parallel plate air capacitor has a capacitance of pf. The plates are at a distance d apart. A slab of thickness t (t < d) and dielectric constant 5 is introduced between the parallel plates. Then the capacitance can be 5 pf pf pf 5 pf 6. A capacitor C is charged to a p.d V. The charging battery is then removed and the capacitor is connected to an uncharged capacitor C. Let U is the initial stored energy in capacitor C, then final stored energy in the combination is C U C C C C U C C C U C C U C C Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

22 PCE 7. A capacitor of µf withstands a maximum voltage of 6 kilovolt while another capacitor of µf withstands a maximum voltage of 4 kilovolt. If the two capacitors are connected in series, the system will withstand a maximum voltage of kv 4 kv 6 kv 9 kv 8. Four metallic plates, each having same area, are placed as shown in figure. The distance between the consecutive plates are equal. Alternate plates are connected to points A and B. Let the equivalent capacitance of this system is C. Now the same four metallic plates are arranged as shown in figure. Let the equivalent capacitance of the system is C Then C /C equals to 5/3 3/5 5/ /5 3. A parallel plate capacitor is connected to a battery. The quantities charge, voltage, electric field and energy associated with this capacitor are given by Q, V, E and U respectively. A dielectric slab is now introduced to fill the space between the plates with the battery still in connection. The corresponding quantities now given by Q, V, E and U are related to the previous ones as Q = Q V > V E > E U > U 3. Seven capacitors each of capacitance µf are to be connected in a configuration to obtain an effective capacitance of (/) µf. Which of the following combination(s) will achieve the desired result? 3 : : 3 : : 5 9. A parallel plate capacitor with a slab of dielectric constant 3 filling the whole space between the plates is charged to a certain potential and isolated. Then the slab is drawn out and another slab of equal thickness but dielectric constant is introduced between the plates. The ratio of the energy stored in the capacitor later to that shored initially is : 3 3 : 4 : 9 9 : 4 3. The figure shows two identical parallel plate capacitors connected to a battery with switch S closed. The switch is now opened and the free space between the plates of the capacitors is filled with a dielectric of constant 3. The ratio of the total energy stored in both capacitors before and after the introduction of the dielectric is Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

23 33. A parallel plate capacitor of plate area A and plate separation d, is charged to potential difference V and then the battery is disconnected. A slab of dielectric constant K is then inserted between the plates of the capacitor so as to fill the space between the plates. If Q, E and W denote respectively, the magnitude of charge on each plate, the electric field between the plates (after the slab is inserted), and work done on the system, in question, in the process of inserting the slab, then AV d Q Q E V Kd W KAV d AV d COMPREHENSION TYPE Comprehension- Electric fish are able to generate current with biological cells called electroplaques, which are physiological emf devices. The electroplaques in the South American eel are arranged in 4 rows, each row stretching horizontally along the body and each containing 5 electroplaques. Each electroplaque has an emf of.5 V and an internal resistance r of.5. The water surrounding the eel completes a circuit between the two ends of the electroplque array, one end at the animal s head and the other near its tail.. If the water surrounding the eel has resistance R w = 8, the current can be eel produce in the water K.93 A.83 A.67 A.76 A. The current travels through each row of the figure is.6 3 A A A A PCE The capacities of two conductors are C and C and their respective potentials are V and V. If they are connected by a thin wire, then the loss of energy will be Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857 CC (V V ) (C C ) CC (V V ) (C C ) CC (V V ) (C C ) both and are correct 35. A resistance of is connected across one gap of a meter-bridge (the length of the wire is cm) and an unknown resistance, greater than, is connected across the other gap. When these resistances are interchanged, the balance point shifts by cm. Neglecting any corrections, the unknown resistance is EXCERCISE BASED ON NEW PATTERN Comprehension- Two capacitors A and B with capacities 3µF and µf are charged to a potential difference of V and 8 V respectively. The plates of the capacitors are connected as shown in the figure with one wire from each capacitor free. The upper plate of A is positive and that of B is negative. An uncharged µf capacitor C with lead wires falls on the free ends to complete the circuit calculate : 3. The final charge on the capacitor C is 9 µc 5 µc µc 8 µc 4. The final charge on A is 9 µc 5 µc µc 8 µc 5. The amount of heat energy produced 8 mj mj mj none

24 PCE 4 Comprehension-3 Fuse is a metallic conducting wire of 75% Pb and 5% Sn with low melting point and is put in series with an appliance. It is safety device which protects the appliance from getting damaged, by melting and opening the circuit if the current in the circuit exceeds a specific predetermined value, called current capacity. In a steady state, I R = ha. Here I is the current through the fuse, R is the resistance of the fuse, h is the heat lost per unit time per unit area and A is the surface area. 6. The current capacity of the fuse depends on radius of the wire length of the wire both radius as well as length none of these 7. Consider that radiation losses are neglected and hence temperature of fuse wire will increase continuously and after certain time it will melt. Let this time is directly proportional to r, where r is the radius of the wire and is a constant quantity. The value of is The heating element of an electric heater should be made of a material having Comprehension-4 high specific resistance and high melting point high specific resistance and low melting point low specific resistance and low melting point low specific resistance and high melting point. A parallel-plate capacitor of µf is connected across a source of constant emf of 3 V. Without disconnected the source, a dielectric ( = 4) is introduced to fill the space between the plates. 9. The energy drawn from the source during the introduction of the dielectric is.54 J.7 J.3 J None. The extra energy stored in the capacitor due to introduction of the dielectric is.54 J.7 J.3 J None. The amount of heat energy produced is.54 J.7 J.3 J None Comprehension-5 In the circuit shown in the figure, E = 3V, E = V, E 3 = V and R = r = r = r 3 =.. The potential difference between the points A and B is V V.5 V 3 V 3. The currents through E is.5 A A A 4. If r is short-circuited and the point A is connected to point B then the curent in the resistor R is.5 A A A MATRIX-MATCH TYPE Matching- Column - A Column - B (A) Measuring the emf of (P) Wheat stone the battery bridge (B) Should be zero (Q) Potentiometer resistance (C) Meter bridge (R) Ammeter (D) Post office box (S) Volt-meter Matching- Column - A (T) Column - B Measuring the resistance (A) Energy storage device (P) Capacitor (B) Charging battery (Q) Inductor (C) Discharging battery (R) Potential difference greater than emf Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

25 PCE 5 (D) Passive elements (S) Potential difference less than emf Matching-3 Column-A (T) Column-B Resistor (A) Current density (P) Vector (B) Current (Q) Scalar (C) Drift velocity (R) Different for different conductor (D) Resistivity (S) Depends on the dimension of the conductor Matching-4 Column-A (T) Column-B (A) Drift speed of electron (P) 4 in m/sec (B) Thermal speed of electron (Q) 4 in m/sec (C) Number of free electrons (R) 5 per unit volume in the conductor (D) Electrical resitivity of (S) 8 insulator in -m (E) Electrical resitivity of (T) 8 conductor in -m Matching-5 Depends on the electric field In the circuit shown E, F, G and H are cells of emf,, 3 and V, respectively, and internal resistance,, 3 and, respectively. Column-A Column-B (A) The potential difference (P) /3 V across B and D (B) The potential difference (Q) 4/3 V across cell G (C) The potential difference (R) 8/3 V across cell F (D) The potential difference (S) /3 V across cell H MULTIPLE CORRECT CHOICE TYPE. Two cells of unequal emfs, E and E and internal resistances r and r are joined in parallel as shown. the p.d. across both the cells will be equal one cell will continuosly supply energy to the other the p.d. across one cell will be greater than its emf V AB Er E r r r. The figure shows a potentiometer circuit, in which E is the emf of the driving cell and E is the emf of the cell whose emf is to be determined. AB is the potentiometer wire, G is the galvanometer and J is the sliding contact. Which of the following are the essential conditions for obtaining a balance? the resistance of G must be less than the resistance of wire AB the positive terminals of both E and E must be connected to point A either the positive terminals or the negative terminals of both E and E must be connected to point A E must be greater than E Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857

26 PCE 6 3. In the network shown V AB = 3. V V CB = 6. V I =.5 A I =. A 4. There are two cells X and Y as shown in the circuit. EMF of each cell is V, the internal resistance of cell X is. and cell Y is.. I = A A voltmeter connected across X will read higher than voltmeter across Y A voltmeter Y will read.6 V None of the above 5. For a cell, the graph between the potential difference (V) across the terminals of the cell and the current (I) drawn from the cell is shown in the figure. the emf of the cell is more than V the emf of the cell is V the emf of the cell is less than V the internal resistance of the cell is.4 6. Two capacitors of capacity µf and µf are separately charged by a common battery. The two capacitors discharge through equal resistances at t =. The current in each of the two discharging circuit is zero at t = The currents in the two discharging circuits at t = are equal but not zero The currents in two discharging circuit are equal at all times. Capacitor of µf capacity loses 5% of its charge earlier than capacitor of capacity µf 7. A micrometer has a resistance of and a full-scale range of 5 µa. It can be used as a voltmeter or as a higher range ammeter provided a resistance is added to it. Pick the correct range and resistance combination(s). 5 V range with K resistance in series V range with K resistance in series 5 ma range with resistance in parallel ma range with resistance in parallel 8. A parallel plate capacitor is charged and the charging battery is then disconnected. If the plates of the capacitor are moved further apart by means of insulating handels. the charge on the capacitor increases the voltage across the plate increases the capacitance increases the electrostatic energy stored in the capacitor increases. 9. A parallel plate capacitor is filled with a uniform dielectric. Maximum charge that can be given to it depends upon dielectric constant of the dielectric dielectric strength of the dielectric separation between the plates area of the plates. A parallel plate capacitor of plate area A and plate separation d is charged to potential difference V and then the battery is disconnected. A slab of dielectric constant K is then inserted between the plates of the capacitor so as to fill the space between the plates. If Q, E and W denote respectively, the magnitude of charge on each plate, the electric field between the plates (after the slab is inserted), and work done on the system, in question, in the process of inserting the slab, then Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi 8, Ph. : , 857