Capacitance PYQs Set 10 (PDF)

Set 10 of 11 · Questions 271-300 of 304

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Mere Teacher · Previous Year Questions JEE Main Physics · Capacitance · Set 10 (Q271-300)
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271. $A capacitor has air as dielectric medium and two conducting plates of area 12 cm^{2} and they are 0.6 cm apart. When a slab of dielectric having area 12 cm^{2} and 0.6 cm thickness is inserted between the plates, one of the conducting plates has to be moved by 0.2 cm to keep the capacitance same as in previous case. The dielectric constant of the slab is : (Given ϵ_{0} = 8.834 \times 10^{- 12} F / m)$ (1) $1$ (2) $1.33$ (3) $0.66$ (4) $1.50$ 272. $sectional area as the plates of a parallel plate \frac{3}{4} d capacitor and thickness d, where d is the separation of the plates. The capacitance of the capacitor when the slab is inserted between the plates will be: (Given C_{0} = capacitance of capacitor with air as medium between plates.)$ (1) $\frac{4 K C _{0}}{3 + K}$ (2) $\frac{3 K C _{0}}{3 + K}$ (3) $\frac{3 + K}{4 K C _{0}}$ (4) $\frac{K}{4 + K}$ 273. $A parallel plate capacitor is of area 6 c m^{2} and a separation 3 mm . The gap is filled with three dielectric materials of equal thickness (see figure) with dielectric constants K_{1} = 10, K_{2} = 12 and K_{3} = 14 . The dielectric constant of a material which when fully inserted in above capacitor, gives same capacitance would be:$ (1) $4$ (2) $14$ (3) $12$ (4) $26$ 274. $The equivalent capacitance of the combination shown is$ (1) $2C$ (2) $f C$ (3) $\frac{C}{2}$ (4) $4C$ 275. $An infinitely long thin non-conducting wire is parallel to the z-axis and carries a uniform line charge density λ . It pierces a thin non-conducting spherical shell of radius R in such a way that the arc PQ subtends an angle 12 0^{\circ} at the center O of the spherical shell, as shown in the figure. The permittivity of free space is ϵ_{0} . Which of the following statements is (are) true?$ (1) $The electric flux through the shell is 3 R λ / ϵ_{0}$ (2) $The z-component of the electric field is zero at all the points on the surface of the shell$ (3) $The electric flux through the shell is 2 R λ / ϵ_{0}$ (4) $The electric field is normal to the surface of the shell at all points$ 276. $Seven capacitors, each of capacitance 2µF. are to be connected in a configuration to obtain an effective capacitance of . Which of the combinations, shown in figures below, will achieve the desired value?$ (1) (2) (3) (4) 277. $A force of 10N acts on a charged particle placed between two plates of a charged capacitor. If one plate of capacitor is removed, then the force acting on that particle will be:$ (1) $5N$ (2) $10N$ (3) $20 N$ (4) $Zero$ 278. $A parallel plate capacitor has plates of area A separated by distance d between them. It is filled with a dielectric a dielectric which has a dielectric constant that varies as k (x) = k_{0} (1 + αx) where 'x' is the distance measured from one of the plates. If (αd) << 1, the total capacitance of the system is best given by the expression:$ (1) $\frac{k _{0} ϵ _{0} A}{d} (1 + α^{2} d^{2})$ (2) $\frac{k _{0} ϵ _{0} A}{d} (1 + \frac{α d}{2})$ (3) $\frac{k _{0} ϵ _{0} A}{2 d} (1 + αd)$ (4) $\frac{k _{0} ϵ _{0} A}{2 d} (1 + \frac{αd}{2})$ 279. $A parallel plate capacitor was made with two rectangular plates, each with a length of l 3 cm and breath of b=1 cm. The distance between the plates is 3μ m. Out of the following, which are the ways to increase the capacitance by a factor of 10? (A) l = 30 cm, b = 1 cm, d = 1 μ m (B) l = 3 cm, b = 1 cm, d = 30 μ m (C) l = 6 cm, b = 5 cm, d = 3 μ m (D) l = 1 cm, b = 1 cm, d = 10 μ m (E) l = 5 cm, b = 2 cm, d = 1 μ m Choose the correct answer from the options given below:$ (1) $A only$ (2) $C only$ (3) $B and D only$ (4) $C and E only$ 280. $The electric field E is measured at a point P(0, 0, d) generated due to various charge distributions and the dependence of E on d is found to be different for different charge distributions. List-I contains different relations between E and d. List-II describes different electric charge distributions, along with their locations. Match the functions in List-I with the related charge distributions in List-II. List-I P. E is independent of d Q. E μ 1/d R. E μ 1/ d^{2} S. E μ 1/ d^{3} List-II 1. A point charge Q at the origin 2. A small dipole with point charges Q at (0, 0, l) and -Q at (0, 0, -l) Take 2l << d 3. An infinite line charge coincident with the x-axis, with uniform linear charge density l. 4. Two infinite wires carrying uniform linear charge density parallel to the x-axis. The one along (y = 0, z = l) has a charge density +l and the one along (y = 0, z = -l) has a charge density -l. Take 2l << d 5. Infinite plane charge coincident with the xy-plane with uniform surface charge density.$ (1) $P \to 5, Q \to 3, 4, R \to 1, S \to 2$ (2) $P \to 5, Q \to 3, R \to 1, 4, S \to 2$ (3) $P \to 5, Q \to 3, R \to 1, 2, S \to 4$ (4) $P \to 4, Q \to 2, 3, R \to 1, S \to 5$ 281. $Four charge Q_{1}, Q_{2}, Q_{3}, and Q_{4}, of same magnitude are fixed along the x axis at x = -2a -a, +a and +2a, respectively. A positive charge q is placed on the positive y axis at a distance b > 0. Four options of the signs of these charges are given in List-I. The direction of the forces on the charge q is given in List- II Match List-1 with List-II and select the correct answer using the code given below the lists. List-I (P) Q_{1}, Q_{2}, Q_{3}, Q_{4}, all positive (Q) Q_{1}, Q_{2} positive Q_{3}, Q_{4} negative (R) Q_{1}, Q_{4} positive Q_{2}, Q_{3} negative (S) Q_{1}, Q_{3} positive Q_{2}, Q_{4} negative List-II (1) +x (2) -x (3) +y (4) -y$ (1) $P-3, Q-1, R-4, S-2$ (2) $P-4, Q-2, R-3, S-1$ (3) $P-3, Q-1, R-2, S-4$ (4) $P-4, Q-2, R-1, S-3$ 282. $This question has statement-I and statement-II. Of the four choices given after the statements, choose the one that best describes the two statements. An insulating solid sphere of radius R has a uniformly positive charge density ρ. As a result of this uniform charge distribution there is a finite value of electric potential at the center of the sphere, at the surface of the sphere and also at a point out side the sphere. The electric potential at infinite is zero. Statement-I: When a charge ‘q’ is taken from the center to the surface of the sphere its potential energy changes by \frac{q ρ}{3 ϵ _{0}} . Statement-II: The electric field at a distance r (r < R) from the center of the sphere is \frac{ρ r}{3 ϵ _{0}}$ (1) $Statement-I is true, Statement-II is true; Statement-II is not the correct explanation of statement-I.$ (2) $Statement I is true Statement II is false.$ (3) $Statement I is false Statement II is true.$ (4) $Statement I is true, Statement II is true, Statement II is the correct explanation of Statement I.$ 283. $A parallel plate capacitor has a dielectric slab of dielectric constant K between its plates that covers 1/3 of the area of its plates, as shown in the figure. The total capacitance of the capacitor is C while that of the portion with dielectric in between is C_{1} . When the capacitor is charged, the plate area covered by the dielectric gets charge Q_{1} and the rest of the area gets charge Q_{2} . Choose the correct option/options, ignoring edge effects.$ (1) $\frac{E _{1}}{E _{2}} = 1$ (2) $\frac{E _{1}}{E _{2}} = \frac{1}{K}$ (3) $\frac{Q _{1}}{Q _{2}} = \frac{3}{K}$ (4) $\frac{C}{C _{1}} = \frac{2 + K}{K}$ 284. $Two identical parallel plate capacitors, of capacitance C each, have plates of area A, separated by a distance d . The space between the plates of the two capacitors, is filled with three dielectrics, of equal thickness and dielectric constants K_{1}, K_{2} and K_{3} . The first capacitors is filled as shown in Fig. I, and the second one is filled as shown in Fig. II. If these two modified capacitors are charged by the same potential V, the ratio of the energy stored in the two would be (E_{1} refers to capacitors (I) and E_{2} to capacitors (II):$ (1) $\frac{E _{1}}{E _{2}} = \frac{K _{1} K _{2} K _{3}}{( K _{1} + K _{2} + K _{3} ) ( K _{2} K _{3} + K _{3} K _{1} + K _{1} K _{2} )}$ (2) $\frac{E _{1}}{E _{2}} = \frac{( K _{1} + K _{2} + K _{3} ) ( K _{2} K _{3} + K _{3} K _{1} + K _{1} K _{2} )}{K _{1} K _{2} K _{3}}$ (3) $\frac{E _{1}}{E _{2}} = \frac{9 K _{1} K _{2} K _{3}}{( K _{1} + K _{2} + K _{3} ) ( K _{2} K _{3} + K _{3} K _{1} + K _{1} K _{2} )}$ (4) $\frac{E _{1}}{E _{2}} = \frac{( K _{1} + K _{2} + K _{3} ) ( K _{2} K _{3} + K _{3} K _{1} + K _{1} K _{2} )}{9 K _{1} K _{2} K _{3}}$ 285. $A capacitor, C_{1} = 6 μ F is charged to a potential difference of V_{0} = 5 V using a 5 V battery. The battery is removed and another capacitor, C_{2} = 12 μ F is inserted in place of the battery. When the switch 'S' is closed, the charge flows between the capacitors for some time until equilibrium condition is reached. What are the charges (q_{1} and q_{2}) on the capacitors C_{1} and C_{2} when equilibrium condition is reached.$ (1) $q_{1} = 10 μC, q_{2} = 20 μC$ (2) $q_{1} = 30 μC, q_{2} = 15 μC$ (3) $q_{1} = 20 μC, q_{2} = 10 μC$ (4) $q_{1} = 15 μC, q_{2} = 30 μC$ 286. $A disk of radius a/4 having a uniformly distributed charge 6C is placed in the x-y plane with its center at (-a/2, 0, 0). A rod of length a carrying a uniformly distributed charge 8C is placed on the x-axis from x = a/4 to x = 5a/4. Two point charges -7C and 3C are placed at (a/4, -a/4, 0) and (-3a/4, 3a/4, 0), respectively. Consider a cubical surface formed by six surfaces x = \pm a/2, y = \pm a/2, z = \pma/2. The electric flux through this cubical surface is:$ (1) $\frac{- 2 C}{ϵ _{0}}$ (2) $\frac{2 C}{ϵ _{0}}$ (3) $\frac{10 C}{ϵ _{0}}$ (4) $\frac{12 C}{ϵ _{0}}$ 287. $What is the effective capacitance between points X and Y?$ (1) $12$ (2) $18 μ F$ (3) $24 μ F$ (4) $6 μ F$ 288. $An electric dipole has a fixed dipole moment p, which makes angle θ with respect to x-axis. When subjected to an electric field E_{1} = E i, it experiences a torque T_{1} = τ k . When subjected to another electric field E_{2} = 3 E_{1} j it experiences a torque T_{2} = - T_{1} . The angle θ is:$ (1) $90°$ (2) $30°$ (3) $45°$ (4) $60°$ 289. $A capacitor is made of two square plates each of side 'a' making a very small angle α between them, as shown in figure. The capacitance will be close to:$ (1) $\frac{ϵ _{0} a ^{2}}{d} (1 - \frac{α a}{2 d})$ (2) $\frac{ϵ _{0} a ^{2}}{d} (1 + \frac{α a}{2 d})$ (3) $\frac{ϵ _{0} a ^{2}}{d} (1 - \frac{α a}{4 d})$ (4) $\frac{ϵ _{0} a ^{2}}{d} (1 - \frac{3 α a}{2 d})$ 290. $In the arrangement of capacitors shown in the figure, if each capacitor is 9 pF, then the effective capacitance between the points A and B is$ (1) $10 pF$ (2) $15 pF$ (3) $20 pF$ (4) $5 pF$ 291. $In the given network, the value of C, so that an equivalent capacitance between A and B is 3 μF, is$ (1) $\frac{1}{5} μF$ (2) $\frac{31}{5} μF$ (3) $48 μF$ (4) $36 μF$ 292. $In the reported figure, a capacitor is formed by placing a compound dielectric (three layers in series) between the plates of a parallel plate capacitor: layer 1 of dielectric constant K and thickness d, layer 2 of dielectric constant 3 K and thickness 2 d, layer 3 of dielectric constant 5 K and thickness 3 d . The expression for the capacity of the said capacitor will be : (Given area of plate = A)$ (1) $\frac{15}{34} \frac{K ϵ _{0} A}{d}$ (2) $\frac{15}{6} \frac{K ϵ _{0} A}{d}$ (3) $\frac{25}{6} \frac{K ϵ _{0} A}{d}$ (4) $\frac{9}{6} \frac{K ϵ _{0} A}{d}$ 293. $The energy stored in a capacitor of capacity C and potential V is given by$ (1) $\frac{CV}{2}$ (2) $\frac{C ^{2} V ^{2}}{2}$ (3) $\frac{C ^{2} V}{2}$ (4) $\frac{C V ^{2}}{2}$ 294. $25 capacitors each of capacitance 1 μF are connected in series to a battery of 100 V . The total charge stored on capacitors is$ (1) $2.0 \times 10^{- 5} C$ (2) $2.5 \times 10^{- 3} C$ (3) $4.0 \times 10^{- 6} C$ (4) $1.5 \times 10^{- 6} C$ 295. $Two parallel plate capacitors C_{1} and C_{2} each having capacitance of 10µF are individually charged by a 100 V D.C. source. Capacitor Q is kept connected to the source and a dielectric slab is inserted between it plates. Capacitor C_{2} is disconnected from the source and then a dielectric slab is inserted in it. Afterwards the capacitor C_{1} is also disconnected from the source and the two capacitors are finally connected in parallel combination. The common potential of the combination will be ________V. (Assuming Dielectric constant = 10)$ 296. $In the given circuit. C_{1} = 2µF, C_{2} = 0.2µF, C_{3} = 2µF, C_{4} = 4µF, C_{5} = 2µF, C_{6} = 2µF. The charge stored on capacitor C_{4} is ______ µC.$ 297. $A 2 μ F capacitor C_{1} is first charged to a potential difference of 10 V using a battery. Then the battery is removed and the capacitor is connected to an uncharged capacitor C_{2} of 8 μ F . The charge in C_{2} on equilibrium condition is _____μ C . (Round off to the Nearest Integer)$ 298. $A 60 pF capacitor is fully charged by a 20 V supply. It is then disconnected from the supply and is connected to another uncharged 60 pF capacitor in parallel. The electrostatic energy that is lost in this process by the time the charge is redistributed between them is (in nJ) ____.$ 299. $The figure shows two identical parallel plate capacitors connected to a battery with the switch S closed. The switch is now opened and the free space between the plates of the capacitors is filled with a dielectric of dielectric constant (or relative permittivity) 3. Find the ratio of the total electrostatic energy stored in both capacitors before and after the introduction of the dielectric.$ 300. $Three identical capacitors C1, C_{2} and C3 have a capacitance of 1.0 µF each and they are uncharged initially. They are connected in a circuit as shown in the figure and C1 is then filled completely with a dielectric material of relative permittivity εr. The cell electromotive force (emf) V_{0} = 8V. First the switch S1 is closed while the switch S2 is kept open. When the capacitor C3 is fully charged, S1 is opened and S2 is closed simultaneously. When all the capacitors reach equilibrium, the charge on C3 is found to be 5 µC. The value of εr =____________.$