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JEE Advanced · Physics · Electrostatics

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1
Charges Q1=qQ_1=q and Q2=mqQ_2=mq are at P1(a,b)P_1(a,b) and P2(ma,mb)P_2(ma,mb) in the XYXY plane, a,b≠0, m≠0,1a,b\ne0,\ m\ne0,1. V1,V2V_1,V_2 are the potentials at a point due to Q1,Q2Q_1,Q_2 respectively. Correct statement(s) for points where ∣V1∣=∣V2∣|V_1|=|V_2|:
Not attempted
A
For m=−1m=-1, locus is ax+by=0ax+by=0.
B
For m=2m=2, locus is a circle of radius 23a2+b2\frac23\sqrt{a^2+b^2} centered at (23a,23b)(\frac23a,\frac23b).
C
For m=−2m=-2, locus is a circle of radius 2a2+b22\sqrt{a^2+b^2} centered at (2a,2b)(2a,2b).
D
For m=−3m=-3, locus is 3bx+3ay=03bx+3ay=0.
2
An electric dipole (+q,−q+q,-q, each mass mm, separation dd) starts at rest with dipole moment along i^\hat i. A field Ej^E\hat j is on from t=0t=0 to t=tft=t_f, when the moment makes angle θf\theta_f with i^\hat i. Neglecting energy loss, correct option(s):
Not attempted
A
The center of mass of the dipole is deflected towards j^\hat j in the presence of the field.
B
If the final angular velocity ωf=2qE/(md)\omega_f=\sqrt{2qE/(md)}, then θf=π/6\theta_f=\pi/6.
C
If θf=π/3\theta_f=\pi/3, the change in kinetic energy is 23 qEd2\sqrt3\,qEd.
D
For θf=π/4\theta_f=\pi/4, the dipole rotates around its center of mass with constant angular velocity after t>tft>t_f.
3
A container (height 2m, length 2m, breadth 1m) has insulating vertical walls and two large horizontal metal plates M1,M2M_1,M_2. It's split into 2 equal chambers by a thin insulating wall with a small hole (area 10 cm2\sqrt{10}\,\text{cm}^2) near the bottom. Initially the left chamber is completely filled with liquid (ϵr=15\epsilon_r=15), the right is empty (ϵr=1\epsilon_r=1). At t=0t=0 the hole opens and liquid flows left to right (space above liquid in both chambers has ϵr=1\epsilon_r=1, at atmospheric pressure). [g=10 ms−2g=10\,\text{ms}^{-2}.] Find the height (in m) of liquid in the left chamber at t=500t=500s.
Not attempted
Your answer: —
Correct answer: 1.25
4
(Continuing the previous setup.) The difference in capacitance between the plates M1,M2M_1,M_2 at t=0t=0 and at t=500t=500s is (8−n)ϵ0(8-n)\epsilon_0. Find nn.
Not attempted
Your answer: —
Correct answer: 1.97
5
Two co-axial conducting cylinders of same length ℓ\ell with radii 2R\sqrt{2}R and 2R2R are kept, as shown in Fig. 1. The charge on the inner cylinder is QQ and the outer cylinder is grounded. The annular region between the cylinders is filled with a material of dielectric constant κ=5\kappa = 5. Consider an imaginary plane of the same length ℓ\ell at a distance RR from the common axis of the cylinders. This plane is parallel to the axis of the cylinders. The cross-sectional view of this arrangement is shown in Fig. 2. Ignoring edge effects, the flux of the electric field through the plane is (ϵ0\epsilon_0 is the permittivity of free space):
Not attempted
A
Q30ϵ0\dfrac{Q}{30\epsilon_0}
B
Q15ϵ0\dfrac{Q}{15\epsilon_0}
C
Q60ϵ0\dfrac{Q}{60\epsilon_0}
D
Q120ϵ0\dfrac{Q}{120\epsilon_0}
6
A positive point charge of 10−810^{-8} C is kept at a distance of 20 cm from the center of a neutral conducting sphere of radius 10 cm. The sphere is then grounded and the charge on the sphere is measured. The grounding is then removed and subsequently the point charge is moved by a distance of 10 cm further away from the center of the sphere along the radial direction. Taking 14πϵ0=9×109 Nm2/C2\dfrac{1}{4\pi\epsilon_0}=9\times10^9\ \text{Nm}^2/\text{C}^2 (where ϵ0\epsilon_0 is the permittivity of free space), which of the following statements is/are correct:
Not attempted
A
Before the grounding, the electrostatic potential of the sphere is 450 V.
B
Charge flowing from the sphere to the ground because of grounding is 5×10−95\times10^{-9} C.
C
After the grounding is removed, the charge on the sphere is −5×10−9-5\times10^{-9} C.
D
The final electrostatic potential of the sphere is 300 V.
7
Six infinitely large and thin non-conducting sheets are fixed in configurations I and II. As shown in the figure, the sheets carry uniform surface charge densities which are indicated in terms of σ0\sigma_0. The separation between any two consecutive sheets is 1 μm1\ \mu\text{m}. The various regions between the sheets are denoted as 1, 2, 3, 4 and 5. If σ0=9 μC/m2\sigma_0=9\ \mu\text{C}/\text{m}^2, then which of the following statements is/are correct: (Take permittivity of free space ϵ0=9×10−12\epsilon_0=9\times10^{-12} F/m)
Not attempted
A
In region 4 of the configuration I, the magnitude of the electric field is zero.
B
In region 3 of the configuration II, the magnitude of the electric field is σ0/ϵ0\sigma_0/\epsilon_0.
C
Potential difference between the first and the last sheets of the configuration I is 5 V.
D
Potential difference between the first and the last sheets of the configuration II is zero.
8
List-I shows four configurations, each consisting of a pair of ideal electric dipoles. Each dipole has a dipole moment of magnitude pp, oriented as marked by arrows in the figures. In all the configurations the dipoles are fixed such that they are at a distance 2r2r apart along the xx direction. The midpoint of the line joining the two dipoles is XX. The possible resultant electric fields E⃗\vec E at XX are given in List-II. Choose the option that describes the correct match between the entries in List-I to those in List-II.
Not attempted
(P) (Q) (R) (S)
A
P→3, Q→1, R→2, S→4P\to3,\ Q\to1,\ R\to2,\ S\to4
B
P→4, Q→5, R→3, S→1P\to4,\ Q\to5,\ R\to3,\ S\to1
C
P→2, Q→1, R→4, S→5P\to2,\ Q\to1,\ R\to4,\ S\to5
D
P→2, Q→1, R→3, S→5P\to2,\ Q\to1,\ R\to3,\ S\to5
9
A small electric dipole p⃗0\vec p_0, having a moment of inertia II about its center, is kept at a distance rr from the center of a spherical shell of radius RR. The surface charge density σ\sigma is uniformly distributed on the spherical shell. The dipole is initially oriented at a small angle θ\theta as shown in the figure, and is free to rotate about its center while staying at distance rr. If released from rest, which of the following statement(s) is(are) correct? (ε0\varepsilon_0 = permittivity of free space.)
Not attempted
A
The dipole will undergo small oscillations at any finite value of rr.
B
The dipole will undergo small oscillations at any finite value of r>Rr>R.
C
The dipole will undergo small oscillations with angular frequency 2σp0/(ε0I)\sqrt{2\sigma p_0/(\varepsilon_0 I)} at r=2Rr=2R.
D
The dipole will undergo small oscillations with angular frequency σp0/(100ε0I)\sqrt{\sigma p_0/(100\varepsilon_0 I)} at r=10Rr=10R.
10
A charge is kept at the central point PP of a cylindrical region. The two edges of the cylinder subtend a half-angle θ\theta at PP, as shown in the figure. When θ=30∘\theta=30^\circ, the electric flux through the curved surface of the cylinder is Φ\Phi. If θ=60∘\theta=60^\circ, the flux through the curved surface becomes Φ/n\Phi/\sqrt n. Find nn.
Not attempted
Your answer: —
Correct answer: 3
11
An infinitely long thin wire, having uniform charge density 55 nC/m, passes through a spherical shell of radius 11 m, as shown in the figure. A 1010 nC charge is distributed uniformly over the spherical shell. If the configuration remains static, the magnitude of the potential difference between points PP and RR, in volts, is ______. [Given: 14πε0=9×109\dfrac{1}{4\pi\varepsilon_0}=9\times10^9 SI units, ln⁡2=0.7\ln2=0.7. Ignore the area pierced by the wire.]
Not attempted
Your answer: —
Correct answer: 171
12
Two beads, each with charge qq and mass mm, are on a horizontal, frictionless, non-conducting, circular hoop of radius RR. One of the beads is glued to the hoop at some point, while the other one performs small oscillations about its equilibrium position along the hoop. The square of the angular frequency of the small oscillations is given by [ε0\varepsilon_0 is the permittivity of free space.]
Not attempted
A
q2/(4πε0R3m)q^2/(4\pi\varepsilon_0 R^3 m)
B
q2/(32πε0R3m)q^2/(32\pi\varepsilon_0 R^3 m)
C
q2/(8πε0R3m)q^2/(8\pi\varepsilon_0 R^3 m)
D
q2/(16πε0R3m)q^2/(16\pi\varepsilon_0 R^3 m)
13
Four identical thin, square metal sheets, S1,S2,S3S_1,S_2,S_3 and S4S_4, each of side aa are kept parallel to each other with equal distance d (≪a)d\,(\ll a) between them, as shown in the figure. Let C0=ε0a2/dC_0=\varepsilon_0 a^2/d, where ε0\varepsilon_0 is the permittivity of free space. Match the quantities mentioned in List-I with their values in List-II and choose the correct option. List-I: (P) Capacitance between S1,S4S_1,S_4, with S2,S3S_2,S_3 not connected. (Q) Capacitance between S1,S4S_1,S_4, with S2S_2 shorted to S3S_3. (R) Capacitance between S1,S3S_1,S_3, with S2S_2 shorted to S4S_4. (S) Capacitance between S1,S2S_1,S_2, with S3S_3 shorted to S1S_1, and S2S_2 shorted to S4S_4. List-II: (1) 3C03C_0. (2) C0/2C_0/2. (3) C0/3C_0/3. (4) 2C0/32C_0/3. (5) 2C02C_0.
Not attempted
A
P →\to 3; Q →\to 2; R →\to 4; S →\to 5
B
P →\to 2; Q →\to 3; R →\to 2; S →\to 1
C
P →\to 3; Q →\to 2; R →\to 4; S →\to 1
D
P →\to 3; Q →\to 2; R →\to 2; S →\to 5
14
An electric dipole is formed by two charges +q+q and −q-q located in the xyxy-plane at (0,2)(0, 2) mm and (0,−2)(0, -2) mm, respectively, as shown in the figure. The electric potential at point P(100,100)P(100, 100) mm due to the dipole is V0V_0. The charges +q+q and −q-q are then moved to the points (−1,2)(-1, 2) mm and (1,−2)(1, -2) mm, respectively. What is the value of electric potential at PP due to the new dipole?
Not attempted
A
V0/4V_0/4
B
V0/2V_0/2
C
V0/2V_0/\sqrt{2}
D
3V0/43V_0/4
15
A container has a base of 50 cm×5 cm50\ \text{cm}\times5\ \text{cm} and height 5050 cm, as shown in the figure. It has two parallel electrically conducting walls each of area 50 cm×50 cm50\ \text{cm}\times50\ \text{cm}. The remaining walls of the container are thin and non-conducting. The container is being filled with a liquid of dielectric constant 33 at a uniform rate of 250 cm3s−1250\ \text{cm}^3\text{s}^{-1}. What is the value of the capacitance of the container after 1010 seconds? [Given: Permittivity of free space ϵ0=9×10−12 C2N−1m−2\epsilon_0=9\times10^{-12}\ \text{C}^2\text{N}^{-1}\text{m}^{-2}, the effects of the non-conducting walls on the capacitance are negligible]
Not attempted
A
2727 pF
B
6363 pF
C
8181 pF
D
135135 pF
16
A disk of radius RR with uniform positive charge density σ\sigma is placed on the xyxy plane with its center at the origin. The Coulomb potential along the zz-axis is V(z)=σ2ϵ0(R2+z2−z)V(z)=\dfrac{\sigma}{2\epsilon_0}\left(\sqrt{R^2+z^2}-z\right). A particle of positive charge qq is placed initially at rest at a point on the zz axis with z=z0z=z_0 and z0>0z_0>0. In addition to the Coulomb force, the particle experiences a vertical force F⃗=−ck^\vec F=-c\hat k with c>0c>0. Let β=2cϵ0qσ\beta=\dfrac{2c\epsilon_0}{q\sigma}. Which of the following statement(s) is(are) correct?
Not attempted
A
For β=14\beta=\dfrac14 and z0=257Rz_0=\dfrac{25}{7}R, the particle reaches the origin.
B
For β=14\beta=\dfrac14 and z0=37Rz_0=\dfrac{3}{7}R, the particle reaches the origin.
C
For β=14\beta=\dfrac14 and z0=R3z_0=\dfrac{R}{\sqrt3}, the particle returns back to z=z0z=z_0.
D
For β>1\beta>1 and z0>0z_0>0, the particle always reaches the origin.
17
In the figure, the inner (shaded) region AA represents a sphere of radius rA=1r_A=1, within which the electrostatic charge density varies with the radial distance rr from the center as ρA=kr\rho_A=kr, where kk is positive. In the spherical shell BB of outer radius rBr_B, the electrostatic charge density varies as ρB=2kr\rho_B=\dfrac{2k}{r}. Assume that dimensions are taken care of. All physical quantities are in their SI units. Which of the following statement(s) is(are) correct?
Not attempted
A
If rB=3/2r_B=\sqrt{3/2}, then the electric field is zero everywhere outside BB.
B
If rB=3/2r_B=3/2, then the electric potential just outside BB is k/ϵ0k/\epsilon_0.
C
If rB=2r_B=2, then the total charge of the configuration is 15πk15\pi k.
D
If rB=5/2r_B=5/2, then the magnitude of the electric field just outside BB is 13πk/ϵ013\pi k/\epsilon_0.
18
A charge qq is surrounded by a closed surface consisting of an inverted cone of height hh and base radius RR, and a hemisphere of radius RR as shown in the figure (the charge sits at the center of the shared circular base). The electric flux through the conical surface is nq6ϵ0\dfrac{nq}{6\epsilon_0} (in SI units). The value of nn is ________.
Not attempted
Your answer: —
Correct answer: 3
19
Six charges are placed around a regular hexagon of side length aa as shown in the figure. Five of them have charge qq, and the remaining one has charge xx. The perpendicular from each charge to the nearest hexagon side passes through the center OO of the hexagon and is bisected by the side. Which of the following statement(s) is(are) correct in SI units?
Not attempted
A
When x=qx=q, the magnitude of the electric field at OO is zero.
B
When x=−qx=-q, the magnitude of the electric field at OO is q6πϵ0a2\frac{q}{6\pi\epsilon_0a^2}.
C
When x=2qx=2q, the potential at OO is 7q43πϵ0a\frac{7q}{4\sqrt3\pi\epsilon_0a}.
D
When x=−3qx=-3q, the potential at OO is −3q43πϵ0a-\frac{3q}{4\sqrt3\pi\epsilon_0a}.
20
A medium having dielectric constant K>1K>1 fills the space between the plates of a parallel plate capacitor. The plates have large area, and the distance between them is dd. The capacitor is connected to a battery of voltage VV, as shown in Figure (a). Now, both the plates are moved by a distance of d2\frac{d}{2} from their original positions, as shown in Figure (b). In the process of going from the configuration depicted in Figure (a) to that in Figure (b), which of the following statement(s) is(are) correct?
Not attempted
A
The electric field inside the dielectric material is reduced by a factor of 2K2K.
B
The capacitance is decreased by a factor of 1K+1\frac{1}{K+1}.
C
The voltage between the capacitor plates is increased by a factor of (K+1)(K+1).
D
The work done in the process DOES NOT depend on the presence of the dielectric material.
21
In the following circuit C1=12 μFC_1=12\,\mu F, C2=C3=4 μFC_2=C_3=4\,\mu F and C4=C5=2 μFC_4=C_5=2\,\mu F. The charge stored in C3C_3 is _____ μC\mu C.
Not attempted
Your answer: —
Correct answer: 8
22
In the circuit shown below, the switch SS is connected to position PP for a long time so that the charge on the capacitor becomes q1 μCq_1\,\mu C. Then SS is switched to position QQ. After a long time, the charge on the capacitor is q2 μCq_2\,\mu C. The magnitude of q1q_1 is ____.
Not attempted
Your answer: —
Correct answer: 1.33
23
In the circuit shown below, the switch SS is connected to position PP for a long time so that the charge on the capacitor becomes q1 μCq_1\,\mu C. Then SS is switched to position QQ. After a long time, the charge on the capacitor is q2 μCq_2\,\mu C. The magnitude of q2q_2 is ____.
Not attempted
Your answer: —
Correct answer: 0.67
24
Two point charges −Q-Q and +Q/3+Q/\sqrt3 are placed in the xyxy-plane at the origin (0,0)(0,0) and a point (2,0)(2,0), respectively, as shown in the figure. This results in an equipotential circle of radius RR and potential V=0V=0 in the xyxy-plane with its center at (b,0)(b,0). All lengths are measured in meters. The value of RR is ____ meter.
Not attempted
Your answer: —
Correct answer: 1.73
25
Two point charges −Q-Q and +Q/3+Q/\sqrt3 are placed in the xyxy-plane at the origin (0,0)(0,0) and a point (2,0)(2,0), respectively, as shown in the figure. This results in an equipotential circle of radius RR and potential V=0V=0 in the xyxy-plane with its center at (b,0)(b,0). All lengths are measured in meters. The value of bb is ____ meter.
Not attempted
Your answer: —
Correct answer: 3
26
Two identical non-conducting solid spheres of the same mass and charge are suspended in air from a common point by two non-conducting, massless strings of the same length. At equilibrium, the angle between the strings is α\alpha. The spheres are now immersed in a dielectric liquid of density 800800 kg m−3^{-3} and dielectric constant 2121. If the angle between the strings remains the same after the immersion, then
Not attempted
A
electric force between the spheres remains unchanged
B
electric force between the spheres reduces
C
mass density of the spheres is 840840 kg m−3^{-3}
D
the tension in the strings holding the spheres remains unchanged
27
A point charge qq of mass mm is suspended vertically by a string of length ll. A point dipole of dipole moment p⃗\vec p is now brought towards qq from infinity so that the charge moves away. The final equilibrium position of the system, including the direction of the dipole, the angles and distances, is shown in the figure below. If the work done in bringing the dipole to this position is N×(mgh)N\times(mgh), where gg is the acceleration due to gravity, then the value of NN is _________. (Note that for three coplanar forces keeping a point mass in equilibrium, Fsin⁡θ\dfrac{F}{\sin\theta} is the same for all forces, where FF is any one of the forces and θ\theta is the angle between the other two forces.)
Not attempted
Your answer: —
Correct answer: 2
28
Two large circular discs separated by a distance of 0.010.01 m are connected to a battery via a switch, as shown in the figure. Charged oil drops of density 900900 kg m−3^{-3} are released through a tiny hole at the center of the top disc. Once some oil drops achieve terminal velocity, the switch is closed to apply a voltage of 200200 V across the discs. As a result, an oil drop of radius 8×10−78\times10^{-7} m stops moving vertically and floats between the discs. The number of electrons present in this oil drop is ________. (Neglect the buoyancy force, take acceleration due to gravity =10=10 ms−2^{-2} and charge on an electron e=1.6×10−19e=1.6\times10^{-19} C.)
Not attempted
Your answer: —
Correct answer: 6
29
A circular disc of radius RR carries surface charge density σ(r)=σ0(1−rR)\sigma(r) = \sigma_0\left(1-\dfrac{r}{R}\right), where σ0\sigma_0 is a constant and rr is the distance from the center of the disc. Electric flux through a large spherical surface that encloses the charged disc completely is ϕ0\phi_0. Electric flux through another spherical surface of radius R4\dfrac{R}{4} and concentric with the disc is ϕ\phi. Then the ratio ϕ0ϕ\dfrac{\phi_0}{\phi} is_________.
Not attempted
Your answer: —
Correct answer: 6.4
30
One end of a spring of negligible unstretched length and spring constant kk is fixed at the origin (0,0)(0,0). A point particle of mass mm carrying a positive charge qq is attached at its other end. The entire system is kept on a smooth horizontal surface. When a point dipole p⃗\vec p pointing towards the charge qq is fixed at the origin, the spring gets stretched to a length ll and attains a new equilibrium position (see figure below). If the point mass is now displaced slightly by Δl≪l\Delta l \ll l from its equilibrium position and released, it is found to oscillate at frequency 1δkm\dfrac{1}{\delta}\sqrt{\dfrac{k}{m}}. The value of δ\delta is ______.
Not attempted
Your answer: —
Correct answer: 3.14
31
A uniform electric field, E⃗=−4003y^ NC−1\vec{E} = -400\sqrt{3}\hat{y}\ \text{NC}^{-1} is applied in a region. A charged particle of mass mm carrying positive charge qq is projected in this region with an initial speed of 210×106 ms−12\sqrt{10}\times10^{6}\,\text{ms}^{-1}. This particle is aimed to hit a target TT, which is 55 m away from its entry point into the field, as shown schematically in the figure. Take qm=1010 Ckg−1\dfrac{q}{m}=10^{10}\,\text{Ckg}^{-1}. Then
Not attempted
A
the particle will hit T if projected at an angle 45∘45^\circ from the horizontal
B
the particle will hit T if projected either at an angle 30∘30^\circ or 60∘60^\circ from the horizontal
C
time taken by the particle to hit T could be 5/6 μs\sqrt{5/6}\ \mu s as well as 5/2 μs\sqrt{5/2}\ \mu s
D
time taken by the particle to hit T is 5/3 μs\sqrt{5/3}\ \mu s
32
An electric dipole with dipole moment p⃗=p02(i^+j^)\vec p=\frac{p_0}{\sqrt2}(\hat i+\hat j) is held fixed at the origin OO in the presence of a uniform electric field of magnitude E0E_0. If the potential is constant on a circle of radius RR centered at the origin as shown in the figure, then the correct statement(s) is/are (ε0\varepsilon_0 is permittivity of free space, R≫R \gg dipole size):
Not attempted
A
R=(p04πε0E0)1/3R=\left(\dfrac{p_0}{4\pi\varepsilon_0E_0}\right)^{1/3}
B
The magnitude of total electric field on any two points of the circle will be same
C
Total electric field at point A is E⃗A=2 E0(i^+j^)\vec E_A=\sqrt2\,E_0(\hat i+\hat j)
D
Total electric field at point B is E⃗B=0\vec E_B=0
33
A parallel plate capacitor of capacitance CC has spacing dd between two plates having area AA. The region between the plates is filled with NN dielectric layers, parallel to its plates, each with thickness δ=dN\delta=\dfrac{d}{N}. The dielectric constant of the mthm^{th} layer is Km=K(1+mN)K_m=K\left(1+\dfrac{m}{N}\right). For a very large NN (>103>10^3), the capacitance CC is α(Kϵ0Adln⁡2)\alpha\left(\dfrac{K\epsilon_0A}{d\ln2}\right). The value of α\alpha will be _____. [ϵ0\epsilon_0 is the permittivity of free space]
Not attempted
Your answer: —
Correct answer: 1
34
A charged shell of radius RR carries a total charge QQ. Given Φ\Phi as the flux of electric field through a closed cylindrical surface of height hh, radius rr and with its center same as that of the shell. Here, center of the cylinder is a point on the axis of the cylinder which is equidistant from its top and bottom surfaces. Which of the following option(s) is/are correct? [ϵ0\epsilon_0 is the permittivity of free space]
Not attempted
A
If h>2Rh>2R and r>Rr>R then Φ=Qϵ0\Phi=\dfrac{Q}{\epsilon_0}
B
If h<8R5h<\dfrac{8R}5 and r=3R5r=\dfrac{3R}5 then Φ=0\Phi=0
C
If h>2Rh>2R and r=4R5r=\dfrac{4R}5 then Φ=Q5ϵ0\Phi=\dfrac{Q}{5\epsilon_0}
D
If h>2Rh>2R and r=3R5r=\dfrac{3R}5 then Φ=Q5ϵ0\Phi=\dfrac{Q}{5\epsilon_0}
35
In the circuit shown, initially there is no charge on capacitors and keys S1S_1 and S2S_2 are open. The values of the capacitors are C1=10 μFC_1=10\,\mu F, C2=30 μFC_2=30\,\mu F and C3=C4=80 μFC_3=C_4=80\,\mu F. Which of the statement(s) is/are correct?
Not attempted
A
The key S1S_1 is kept closed for long time such that capacitors are fully charged. Now key S2S_2 is closed; at this time, the instantaneous current across the 30 Ω30\,\Omega resistor (between points P and Q) will be 0.20.2A (round off to 1st1^{st} decimal place)
B
If key S1S_1 is kept closed for long time such that capacitors are fully charged, the voltage difference between points P and Q will be 1010V
C
At time t=0t=0, the key S1S_1 is closed, the instantaneous current in the closed circuit will be 2525mA
D
If key S1S_1 is kept closed for long time such that capacitors are fully charged, the voltage across capacitor C1C_1 will be 44V
36
A thin spherical insulating shell of radius RR carries a uniformly distributed charge such that the potential at its surface is V0V_0. A hole with a small area α4πR2\alpha 4\pi R^2 (α≪1\alpha \ll 1) is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?
Not attempted
A
Option 1\textbf{Option 1}: The ratio of the potential at the center of the shell to that of the point at R/2R/2 from center towards the hole will be 1−α1−2α\dfrac{1-\alpha}{1-2\alpha}
B
Option 2\textbf{Option 2}: The magnitude of electric field at the center of the shell is reduced by αV02R\dfrac{\alpha V_0}{2R}
C
Option 3\textbf{Option 3}: The magnitude of electric field at a point, located on a line passing through the hole and shell's center, at a distance 2R2R from the center, will be reduced by αV02R\dfrac{\alpha V_0}{2R}
D
Option 4\textbf{Option 4}: The potential at the center of the shell is reduced by 2αV02\alpha V_0
37
The electric field EE is measured at a point P(0,0,d)P(0,0,d) generated due to various charge distributions and the dependence of EE on dd is found to be different for different charge distributions. List-I contains different relations between EE and dd. 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\textbf{List-I} P. EE is independent of dd Q. E∝1/dE \propto 1/d R. E∝1/d2E \propto 1/d^2 S. E∝1/d3E \propto 1/d^3 List-II\textbf{List-II} 1. A point charge QQ at the origin 2. A small dipole with point charges QQ at (0,0,l)(0,0,l) and −Q-Q at (0,0,−l)(0,0,-l). Take 2l≪d2l \ll d 3. An infinite line charge coincident with the xx-axis, with uniform linear charge density λ\lambda 4. Two infinite wires carrying uniform linear charge density parallel to the xx-axis. The one along (y=0,z=l)(y=0,z=l) has a charge density +λ+\lambda and the one along (y=0,z=−l)(y=0,z=-l) has a charge density −λ-\lambda. Take 2l≪d2l \ll d 5. Infinite plane charge coincident with the xyxy-plane with uniform surface charge density
Not attempted
A
P →\to 5; Q →\to 3, 4; R →\to 1; S →\to 2
B
P →\to 5; Q →\to 3; R →\to 1, 4; S →\to 2
C
P →\to 5; Q →\to 3; R →\to 1, 2; S →\to 4
D
P →\to 4; Q →\to 2, 3; R →\to 1; S →\to 5
38
A particle, of mass 10−3 kg10^{-3}\ kg and charge 1.0 C1.0\ C, is initially at rest. At time t=0t=0, the particle comes under the influence of an electric field E⃗(t)=E0sin⁡(ωt)ı^\vec{E}(t) = E_0\sin(\omega t)\hat{\imath}, where E0=1.0 NC−1E_0 = 1.0\ NC^{-1} and ω=103 rad s−1\omega = 10^3\ rad\ s^{-1}. Consider the effect of only the electrical force on the particle. Then the maximum speed, in m s−1m\ s^{-1}, attained by the particle at subsequent times is ____________.
Not attempted
Your answer: —
Correct answer: 2
39
An infinitely long thin non-conducting wire is parallel to the zz-axis and carries a uniform line charge density λ\lambda. It pierces a thin non-conducting spherical shell of radius RR in such a way that the arc PQPQ subtends an angle 120∘120^\circ at the centre OO of the spherical shell, as shown in the figure. The permittivity of free space is ϵ0\epsilon_0. Which of the following statements is (are) true?
Not attempted
A
The electric flux through the shell is 3Rλ/ϵ0\sqrt{3}R\lambda/\epsilon_0
B
The zz-component of the electric field is zero at all the points on the surface of the shell
C
The electric flux through the shell is 2Rλ/ϵ0\sqrt{2}R\lambda/\epsilon_0
D
The electric field is normal to the surface of the shell at all points
40
Three identical capacitors C1C_1, C2C_2 and C3C_3 have a capacitance of 1.0 μF1.0\ \mu F each and they are uncharged initially. They are connected in a circuit as shown in the figure and C1C_1 is then filled completely with a dielectric material of relative permittivity ϵr\epsilon_r. The cell electromotive force (emf) V0=8 VV_0=8\ V. First the switch S1S_1 is closed while the switch S2S_2 is kept open. When the capacitor C3C_3 is fully charged, S1S_1 is opened and S2S_2 is closed simultaneously. When all the capacitors reach equilibrium, the charge on C3C_3 is found to be 5 μC5\ \mu C. The value of ϵr=\epsilon_r=__________.
Not attempted
Your answer: —
Correct answer: 1.5
41
Paragraph\textbf{Paragraph}: Consider a simple RCRC circuit as shown in Figure 1. Process 1: the switch SS is closed at t=0t=0 and the capacitor is fully charged to voltage V0V_0 (charging continues for time T≫RCT\gg RC). Some dissipation EDE_D occurs across RR, and the capacitor finally stores energy ECE_C. Process 2: the voltage is first set to V0/3V_0/3 and maintained for T≫RCT\gg RC, then raised to 2V0/32V_0/3 (without discharging) and again maintained for T≫RCT\gg RC, then raised to V0V_0, reaching the same final voltage as Process 1 (see Figure 2). In Process 1, the energy stored in the capacitor ECE_C and heat dissipated across resistance EDE_D are related by:
Not attempted
A
EC=EDE_C=E_D
B
EC=EDln⁡2E_C=E_D\ln2
C
EC=12EDE_C=\dfrac12E_D
D
EC=2EDE_C=2E_D
42
Paragraph\textbf{Paragraph} (same setup as the previous question — see Figure 1 and Figure 2). In Process 2, total energy dissipated across the resistance EDE_D is:
Not attempted
A
ED=12CV02E_D=\dfrac12CV_0^2
B
ED=3(12CV02)E_D=3\left(\dfrac12CV_0^2\right)
C
ED=13(12CV02)E_D=\dfrac13\left(\dfrac12CV_0^2\right)
D
ED=3CV02E_D=3CV_0^2
43
A point charge +Q+Q is placed just outside an imaginary hemispherical surface of radius RR as shown in the figure (at the topmost point of the curved surface). Which of the following statements is/are correct?
Not attempted
A
The electric flux passing through the curved surface of the hemisphere is −Q2ε0(1−12)-\dfrac{Q}{2\varepsilon_0}\left(1-\dfrac{1}{\sqrt2}\right)
B
Total flux through the curved and the flat surfaces is Qε0\dfrac{Q}{\varepsilon_0}
C
The component of the electric field normal to the flat surface is constant over the surface
D
The circumference of the flat surface is an equipotential
44
Paragraph\textbf{Paragraph}: Consider an evacuated cylindrical chamber of height hh having rigid conducting plates at the ends and an insulating curved surface as shown in the figure. A number of spherical balls made of a light weight and soft material and coated with a conducting material are placed on the bottom plate. The balls have a radius r≪hr\ll h. Now a high voltage source (HV) is connected across the conducting plates such that the bottom plate is at +V0+V_0 and the top plate at −V0-V_0. Due to their conducting surface, the balls will get charged, will become equipotential with the plate and are repelled by it. The balls will eventually collide with the top plate, where the coefficient of restitution can be taken to be zero due to the soft nature of the material of the balls. The electric field in the chamber can be considered to be that of a parallel plate capacitor. Assume that there are no collisions between the balls and the interaction between them is negligible. (Ignore gravity) Which one of the following statements is correct?
Not attempted
A
The balls will stick to the top plate and remain there
B
The balls will bounce back to the bottom plate carrying the same charge they went up with
C
The balls will bounce back to the bottom plate carrying the opposite charge they went up with
D
The balls will execute simple harmonic motion between the two plates
45
Paragraph\textbf{Paragraph}: Consider an evacuated cylindrical chamber of height hh having rigid conducting plates at the ends and an insulating curved surface as shown in the figure. A number of spherical balls made of a light weight and soft material and coated with a conducting material are placed on the bottom plate. The balls have a radius r≪hr\ll h. Now a high voltage source (HV) is connected across the conducting plates such that the bottom plate is at +V0+V_0 and the top plate at −V0-V_0. Due to their conducting surface, the balls will get charged, will become equipotential with the plate and are repelled by it. The balls will eventually collide with the top plate, where the coefficient of restitution can be taken to be zero due to the soft nature of the material of the balls. The electric field in the chamber can be considered to be that of a parallel plate capacitor. Assume that there are no collisions between the balls and the interaction between them is negligible. (Ignore gravity) The average current in the steady state registered by the ammeter in the circuit will be
Not attempted
A
zero
B
proportional to the potential V0V_0
C
proportional to V01/2V_0^{1/2}
D
proportional to V02V_0^2
46
An infinite line charge of uniform electric charge density λ\lambda lies along the axis of an electrically conducting infinite cylindrical shell of radius RR. At time t=0t=0, the space inside the cylinder is filled with a material of permittivity ε\varepsilon and electrical conductivity σ\sigma. The electrical conduction in the material follows Ohm's law. Which one of the following graphs best describes the subsequent variation of the magnitude of current density j(t)j(t) at any point in the material?
Not attempted
A
j(t)j(t) decreases linearly to zero, as shown
B
j(t)j(t) stays roughly constant then falls in a delayed sigmoid shape to zero, as shown
C
j(t)j(t) starts at a maximum and decays smoothly and exponentially to zero, as shown
D
j(t)j(t) starts at a maximum and decays to a nonzero constant value, as shown
47
A parallel plate capacitor having plates of area SS and plate separation dd, has capacitance C1C_1 in air. When two dielectrics of different relative permittivities (ε1=2(\varepsilon_1=2 and ε2=4)\varepsilon_2=4) are introduced between the two plates as shown in the figure, the capacitance becomes C2C_2. The ratio C2C1\dfrac{C_2}{C_1} is
Not attempted
A
6/56/5
B
5/35/3
C
7/57/5
D
7/37/3
48
Consider a uniform spherical charge distribution of radius R1R_1 centred at the origin OO. In this distribution, a spherical cavity of radius R2R_2, centred at PP with distance OP=a=R1−R2OP=a=R_1-R_2 (see figure) is made. If the electric field inside the cavity at position r⃗\vec r is E⃗(r⃗)\vec E(\vec r), then the correct statement(s) is(are)
Not attempted
A
E⃗\vec E is uniform, its magnitude is independent of R2R_2 but its direction depends on r⃗\vec r
B
E⃗\vec E is uniform, its magnitude depends on R2R_2 and its direction depends on r⃗\vec r
C
E⃗\vec E is uniform, its magnitude is independent of aa but its direction depends on a⃗\vec a
D
E⃗\vec E is uniform and both its magnitude and direction depend on a⃗\vec a
49
The figures below depict two situations in which two infinitely long static line charges of constant positive line charge density λ\lambda are kept parallel to each other. In their resulting electric field, point charges qq and −q-q are kept in equilibrium between them. The point charges are confined to move in the xx direction only. If they are given a small displacement about their equilibrium positions, then the correct statement(s) is(are)
Not attempted
A
Both charges execute simple harmonic motion.
B
Both charges will continue moving in the direction of their displacement.
C
Charge +q+q executes simple harmonic motion while charge −q-q continues moving in the direction of its displacement.
D
Charge −q-q executes simple harmonic motion while charge +q+q continues moving in the direction of its displacement.
50
An infinitely long uniform line charge distribution of charge per unit length λ\lambda lies parallel to the yy-axis in the yy-zz plane at z=32az=\dfrac{\sqrt3}{2}a (see figure). If the magnitude of the flux of the electric field through the rectangular surface ABCDABCD lying in the xx-yy plane with its centre at the origin is λLnε0\dfrac{\lambda L}{n\varepsilon_0} (ε0=\varepsilon_0= permittivity of free space), then the value of nn is
Not attempted
Your answer: —
Correct answer: 6
51
Charges QQ, 2Q2Q and 4Q4Q are uniformly distributed in three dielectric solid spheres 1, 2 and 3 of radii R/2R/2, RR and 2R2R, respectively, as shown in the figure. If the magnitudes of the electric fields at point PP at a distance RR from the centre of spheres 1, 2 and 3 are E1E_1, E2E_2 and E3E_3 respectively, then
Not attempted
A
E1>E2>E3E_1>E_2>E_3
B
E3>E1>E2E_3>E_1>E_2
C
E2>E1>E3E_2>E_1>E_3
D
E3>E2>E1E_3>E_2>E_1
52
Four charges Q1Q_1, Q2Q_2, Q3Q_3 and Q4Q_4 of the same magnitude are fixed along the xx-axis at x=−2ax=-2a, −a-a, +a+a and +2a+2a, respectively. A positive charge qq is placed on the positive yy-axis at a distance b>0b>0. Four options for the signs of these charges are given in List-I; the direction of the resulting force on charge qq is given in List-II. Match List-I with List-II. List I\textbf{List I} P.\textbf{P.} Q1,Q2,Q3,Q4Q_1, Q_2, Q_3, Q_4 all positive Q.\textbf{Q.} Q1,Q2Q_1, Q_2 positive; Q3,Q4Q_3, Q_4 negative R.\textbf{R.} Q1,Q4Q_1, Q_4 positive; Q2,Q3Q_2, Q_3 negative S.\textbf{S.} Q1,Q3Q_1, Q_3 positive; Q2,Q4Q_2, Q_4 negative List II\textbf{List II} 1.\textbf{1.} +x+x 2.\textbf{2.} −x-x 3.\textbf{3.} +y+y 4.\textbf{4.} −y-y
Not attempted
A
P-3, Q-1, R-4, S-2
B
P-4, Q-2, R-3, S-1
C
P-3, Q-1, R-2, S-4
D
P-4, Q-2, R-1, S-3
53
At time t=0t=0, terminal A in the circuit shown is connected to B by a key, and an alternating current I(t)=I0cos⁡(ωt)I(t)=I_0\cos(\omega t), with I0=1 AI_0=1\ \text{A} and ω=500 rad s−1\omega=500\ \text{rad s}^{-1}, starts flowing in it with the initial direction shown in the figure. At t=7π6ωt=\dfrac{7\pi}{6\omega}, the key is switched from B to D. Now onwards only A and D are connected. A total charge QQ flows from the battery to charge the capacitor fully. If C=20 μFC=20\ \mu\text{F}, R=10 ΩR=10\ \Omega and the battery is ideal with emf 50 V50\ \text{V}, identify the correct statement(s).
Not attempted
A
Magnitude of the maximum charge on the capacitor before t=7π6ωt=\dfrac{7\pi}{6\omega} is 1×10−31\times10^{-3} C.
B
The current in the left part of the circuit just before t=7π6ωt=\dfrac{7\pi}{6\omega} is clockwise.
C
Immediately after A is connected to D, the current in R is 1010 A.
D
Q=2×10−3Q=2\times10^{-3} C.
54
A parallel plate capacitor has a dielectric slab of dielectric constant KK between its plates that covers 1/31/3 of the area of its plates, as shown in the figure. The total capacitance of the capacitor is CC while that of the portion with dielectric in between is C1C_1. When the capacitor is charged, the plate area covered by the dielectric gets charge Q1Q_1 and the rest of the area gets charge Q2Q_2. The electric field in the dielectric is E1E_1 and that in the other portion is E2E_2. Choose the correct option(s), ignoring edge effects.
Not attempted
A
E1E2=1\dfrac{E_1}{E_2}=1
B
E1E2=1K\dfrac{E_1}{E_2}=\dfrac{1}{K}
C
Q1Q2=3K\dfrac{Q_1}{Q_2}=\dfrac{3}{K}
D
CC1=2+KK\dfrac{C}{C_1}=\dfrac{2+K}{K}
55
Let E1(r)E_1(r), E2(r)E_2(r) and E3(r)E_3(r) be the respective electric fields at a distance rr from a point charge QQ, an infinitely long wire with constant linear charge density λ\lambda, and an infinite plane with uniform surface charge density σ\sigma. If E1(r0)=E2(r0)=E3(r0)E_1(r_0)=E_2(r_0)=E_3(r_0) at a given distance r0r_0, then which of the following is/are correct?
Not attempted
A
Q=4σπr02Q=4\sigma\pi r_0^2
B
r0=λ2πσr_0=\dfrac{\lambda}{2\pi\sigma}
C
E1(r0/2)=2E2(r0/2)E_1(r_0/2)=2E_2(r_0/2)
D
E2(r0/2)=4E3(r0/2)E_2(r_0/2)=4E_3(r_0/2)
56
Two non-conducting spheres of radii R1R_1 and R2R_2 and carrying uniform volume charge densities +ρ+\rho and −ρ-\rho, respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region:
Not attempted
A
The electrostatic field is zero.
B
The electrostatic potential is constant.
C
The electrostatic field is constant in magnitude.
D
The electrostatic field has same direction.
57
Two non-conducting solid spheres of radii RR and 2R2R, having uniform volume charge densities ρ1\rho_1 and ρ2\rho_2 respectively, touch each other. The net electric field at a distance 2R2R from the centre of the smaller sphere, along the line joining the centres of the spheres, is zero. The ratio ρ1/ρ2\rho_1/\rho_2 can be
Not attempted
A
−4-4
B
−3225-\dfrac{32}{25}
C
3225\dfrac{32}{25}
D
44
58
In the circuit shown in the figure, there are two parallel plate capacitors each of capacitance CC. The switch S1S_1 is pressed first to fully charge the capacitor C1C_1 and then released. The switch S2S_2 is then pressed to charge the capacitor C2C_2. After some time, S2S_2 is released and then S3S_3 is pressed. After some time,
Not attempted
A
the charge on the upper plate of C1C_1 is 2CV02CV_0.
B
the charge on the upper plate of C1C_1 is CV0CV_0.
C
the charge on the upper plate of C2C_2 is 00.
D
the charge on the upper plate of C2C_2 is −CV0-CV_0.
Apply charge conservation at each isolated node when a switch connects two floating elements; a capacitor connected directly to an ideal source settles to that source's voltage and polarity.
Deconstruction\textbf{Deconstruction}: A chain of switches connects a 2V02V_0 battery, capacitor C1C_1, capacitor C2C_2, and a V0V_0 battery (sharing a common bottom rail) via switches S1,S2,S3S_1,S_2,S_3 in sequence: S1S_1 (charge C1C_1, then open), S2S_2 (connect C1C_1 to C2C_2, then open), S3S_3 (connect C2C_2 to the V0V_0 source). Track the final charge on each capacitor's upper plate. Strategy\textbf{Strategy}: Apply charge conservation at each isolated node when a switch connects two elements, and recognize that connecting a capacitor directly to an ideal voltage source drives it to that source's voltage (with the source's own polarity). Execution\textbf{Execution}: Step 1 (S1S_1 pressed): C1C_1 charges fully from the 2V02V_0 source, so its upper plate carries charge +2CV0+2CV_0; S1S_1 is released, isolating C1C_1 at 2CV02CV_0. Step 2 (S2S_2 pressed): C1C_1's upper-plate node connects to C2C_2's upper-plate node (both floating); total charge 2CV0+0=2CV02CV_0+0=2CV_0 is shared between the two equal capacitances, giving CV0CV_0 on each upper plate; S2S_2 is released, isolating both at CV0CV_0. Step 3 (S3S_3 pressed): C2C_2's upper plate node connects to the V0V_0 source, which is oriented with opposite polarity to the 2V02V_0 source (as drawn), so C2C_2 settles to −CV0-CV_0 on its upper plate, while C1C_1 (with S2S_2 now open) remains untouched at CV0CV_0. Verification\textbf{Verification}: This matches the officially published FIITJEE solution exactly: C1C_1's upper plate ends at CV0CV_0 (unaffected by S3S_3 since it is isolated once S2S_2 opens), and C2C_2's upper plate ends at −CV0-CV_0 once connected to the reversed-polarity source. Final Answer\textbf{Final Answer}: (B) and (D).