Core Syllabus Module

Electrostatics

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Chapter Elite-20 Challenge Zone

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JEE Main 2019

LEVELJEE Advanced

Let a total charge $2 Q$ be distributed in a sphere of radius $R$ , with the charge density given by $ρ (r) = k r$ , where $r$ is the distance from the centre. Two charges $A$ and $B$ , of $- Q$ each, are placed on diametrically opposite points, at equal distance $a$ , from the centre. If $A$ and $B$ do not experience any force, then

(A)

a = 8^{- 1/4} R

(B)

a = \frac{3 R}{2 ^{1/4}}

(C)

a = 2^{- 1/4} R

(D)

a = R / 3

JEE Advanced 1999

LEVELJEE Advanced

A non-conducting disc of radius $a$ and uniform positive surface charge density $σ$ is placed on the ground with its axis vertical. A particle of mass $m$ and positive charge $q$ is dropped, along the axis of the disc from a height $H$ with zero initial velocity. The particle has $q / m = 4 ε_{0} g / σ$ . (a) Find the value of $H$ if the particle just reaches the disc. (b) Sketch the potential energy of the particle as a function of its height and find its equilibrium position.

JEE Advanced 2000

LEVELJEE Advanced

Four point charges $+ 8 μ C, - 1 μ C, - 1 μ C$ and $+ 8 μ C$ are fixed at the points $- 27/2 m, - 3/2 m, + 3/2 m$ and $+ 27/2 m$ respectively on the $y$ -axis. A particle of mass $6 \times 1 0^{- 4} kg$ and charge $+ 0.1 μ C$ moves along the $x$ -direction. Its speed at $x = + \infty$ is $v_{0}$ . Find the least value of $v_{0}$ for which the particle will cross the origin. Find also the kinetic energy of the particle at the origin. Assume that space is gravity free. $(1/4 π ε_{0} = 9 \times 1 0^{9} Nm^{2} / C^{2})$

JEE Advanced 1993

LEVELJEE Advanced

Two parallel plate capacitors $A$ and $B$ have the same separation $d = 8.85 \times 1 0^{- 4} m$ between the plates. The plate areas of $A$ and $B$ are $0.04 m^{2}$ and $0.02 m^{2}$ respectively. A slab of dielectric constant (relative permittivity) $K = 9$ has dimensions such that it can exactly fill the space between the plates of capacitor $B$ . (a) The dielectric slab is placed inside $A$ as shown in figure (i). $A$ is then charged to a potential difference of $110 V$ . Calculate the capacitance of $A$ and the energy stored in it. (b) The battery is disconnected and then the dielectric slab is removed from $A$ . Find the work done by the external agency in removing the slab from $A$ . (c) The same dielectric slab is now placed inside $B$ , filling it completely. The two capacitors $A$ and $B$ are then connected as shown in figure (iii). Calculate the energy stored in the system.

JEE Advanced 1991

LEVELJEE Advanced

Two fixed charges $- 2 Q$ and $Q$ are located at the points with coordinates $(- 3 a, 0)$ and $(+ 3 a, 0)$ respectively in the $x - y$ plane. (1991) (a) Show that all points in the $x - y$ plane where the electric potential due to the two charges is zero, lie on a circle. Find its radius and the location of its centre. (b) Give the expression $V (x)$ at a general point on the $x$ -axis and sketch the function $V (x)$ on the whole $x$ -axis. (c) If a particle of charge $+ q$ starts form rest at the centre of the circle, show by a short quantitative argument that the particle eventually crosses the circle. Find its speed when it does so.

JEE Advanced 2013

LEVELJEE Advanced

Two non-conducting solid spheres of radii $R$ and $2 R$ , having uniform volume charge densities $ρ_{1}$ and $ρ_{2}$ respectively, touch each other. The net electric field at a distance $2 R$ from the centre of the smaller sphere, along the line joining the centre of the spheres, is zero. The ratio $ρ_{1} / ρ_{2}$ can be

* Multiple Correct Options

(A)

- 4

(B)

- \frac{32}{25}

(C)

\frac{32}{25}

(D)

4

LEVELJEE Advanced

Two capacitors $A$ and $B$ with capacities $3 μ F$ and $2 μ F$ are charged to a potential difference of $100 V$ and $180 V$ respectively. The plates of the capacitors are connected as shown in the figure with one wire of each capacitor free. The upper plate of $A$ is positive and that of $B$ is negative. An uncharged $2 μ F$ capacitor $C$ with lead wires falls on the free ends to complete the circuit. Calculate (a) the final charge on the three capacitors and (b) the amount of electrostatic energy stored in the system before and after completion of the circuit.

JEE Advanced 2025

LEVELJEE Advanced

Two co-axial conducting cylinders of same length $ℓ$ with radii $2 R$ and $2 R$ are kept, as shown in Fig. 1. The charge on the inner cylinder is $Q$ and the outer cylinder is grounded. The annular region between the cylinders is filled with a material of dielectric constant $κ = 5$ . Consider an imaginary plane of the same length $ℓ$ at a distance R 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 ( $ϵ_{0}$ is the permittivity of free space):

(A)

\frac{Q}{30 ϵ _{0}}

(B)

\frac{Q}{15 ϵ _{0}}

(C)

\frac{Q}{60 ϵ _{0}}

(D)

\frac{Q}{120 ϵ _{0}}

JEE Advanced 2020

LEVELJEE Advanced

A point charge $q$ of mass $m$ is suspended vertically by a string of length $l$ . A point dipole of dipole moment $p$ is now brought towards $q$ 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 $N \times (m g h)$ , where $g$ is the acceleration due to gravity, then the value of $N$ is _________. (Note that for three coplanar forces keeping a point mass in equilibrium, $\frac{F}{s i n θ}$ is the same for all forces, where $F$ is any one of the forces and $θ$ is the angle between the other two forces)

JEE Advanced 2022

LEVELJEE Advanced

A disk of radius $R$ with uniform positive charge density $σ$ is placed on the $x y$ plane with its center at the origin. The Coulomb potential along the $z$ -axis is $V (z) = \frac{σ}{2 ϵ _{0}} (R^{2} + z^{2} - z)$ A particle of positive charge $q$ is placed initially at rest at a point on the $z$ axis with $z = z_{0}$ and $z_{0} > 0$ . In addition to the Coulomb force, the particle experiences a vertical force $F = - c \hat{k}$ with $c > 0$ . Let $β = \frac{2 c ϵ _{0}}{q σ}$ . Which of the following statement(s) is(are) correct?

* Multiple Correct Options

(A)

For

β = \frac{1}{4}

and

z_{0} = \frac{25}{7} R

, the particle reaches the origin.

(B)

For

β = \frac{1}{4}

and

z_{0} = \frac{3}{7} R

, the particle reaches the origin.

(C)

For

β = \frac{1}{4}

and

z_{0} = \frac{R}{3}

, the particle returns back to

z = z_{0}

(D)

For

β > 1

and

z_{0} > 0

, the particle always reaches the origin.

JEE Main 2020

LEVELJEE Advanced

Consider a sphere of radius $R$ which carries a uniform charge density $ρ$ . If a sphere of radius $\frac{R}{2}$ is carved out of it, as shown in the figure the ratio $\frac{∣ E _{A} ∣}{∣ E _{B} ∣}$ of magnitude of electric field $E_{A}$ and $E_{B}$ respectively, at points $A$ and $B$ due to the remaining portion is

(A)

\frac{21}{34}

(B)

\frac{18}{54}

(C)

\frac{17}{54}

(D)

\frac{18}{34}

JEE Main 2019

LEVELJEE Advanced

Four point charges $- q, + q, + q$ and $- q$ are placed on Y-axis at $y = - 2 d$ , $y = - d$ , $y = + d$ and $y = + 2 d$ , respectively. The magnitude of the electric field $E$ at a point on the X-axis at $x = D$ , with $D >> d$ , will behave as

(A)

E \propto \frac{1}{D}

(B)

E \propto \frac{1}{D ^{3}}

(C)

E \propto \frac{1}{D ^{2}}

(D)

E \propto \frac{1}{D ^{4}}

JEE Advanced 1996

LEVELJEE Advanced

The capacitance of a parallel plate capacitor with plate area $A$ and separation $d$ , is $C$ . The space between the plates is filled with two wedges of dielectric constants $K_{1}$ and $K_{2}$ respectively (figure). Find the capacitance of the resulting capacitor.

LEVELJEE Advanced

Three particles, each of mass $1 g$ and carrying a charge $q$ , are suspended from a common point by insulated massless strings, each $100 cm$ long. If the particles are in equilibrium and are located at the corners of an equilateral triangle of side length $3 cm$ , calculate the charge $q$ on each particle. (Take $g = 10 m/s^{2}$ ).

JEE Advanced 1992

LEVELJEE Advanced

(a) If the same charge of $Q$ as in part (a) above is given to a spherical conductor of the same radius $R$ , what will be the energy of the system? (b) A charge of $Q$ is uniformly distributed over a spherical volume of radius $R$ . Obtain an expression for the energy of the system. Assume the earth to be a sphere of uniform mass density. Calculate this energy, given the product of the mass and the radius of the earth to be $2.5 \times 1 0^{31} kg-m$ . (c) What will be the corresponding expression for the energy needed to completely disassemble the planet earth against the gravitational pull amongst its constituent particles?

JEE Advanced 2008

LEVELJEE Advanced

Comprehension Passage

The nuclear charge (

Z e

) is non-uniformly distributed within a nucleus of radius

R

. The charge density

ρ (r)

(charge per unit volume) is dependent only on the radial distance

r

from the centre of the nucleus as shown in figure. The electric field is only along the radial direction.

Question 1:

The electric field at $r = R$ is

(A)

independent of

a

(B)

directly proportional to

a

(C)

directly proportional to

a^{2}

(D)

inversely proportional to

a

Question 2:

For $a = 0$ , the value of $d$ (maximum value of $ρ$ as shown in the figure) is

(A)

\frac{3 Z e}{4 π R ^{3}}

(B)

\frac{3 Z e}{π R ^{3}}

(C)

\frac{4 Z e}{3 π R ^{3}}

(D)

\frac{Z e}{3 π R ^{3}}

Question 3:

The electric field within the nucleus is generally observed to be linearly dependent on $r$ . This implies

(A)

a = 0

(B)

a = \frac{R}{2}

(C)

a = R

(D)

a = \frac{2 R}{3}

JEE Advanced 1998

LEVELJEE Advanced

A conducting sphere $S_{1}$ of radius $r$ is attached to an insulating handle. Another conducting sphere $S_{2}$ of radius $R$ is mounted on an insulating stand. $S_{2}$ is initially uncharged. $S_{1}$ is given a charge $Q$ , brought into contact with $S_{2}$ and removed. $S_{1}$ is recharged such that the charge on it is again $Q$ and it is again brought into contact with $S_{2}$ and removed. This procedure is repeated $n$ times. (a) Find the electrostatic energy of $S_{2}$ after $n$ such contacts with $S_{1}$ . (b) What is the limiting value of this energy as $n \to \infty$ ?

LEVELJEE Advanced

Three concentric spherical metallic shells, $A$ , $B$ and $C$ of radii $a$ , $b$ and $c$ ( $a < b < c$ ) have surface charge densities $σ$ , $- σ$ and $σ$ respectively. (a) Find the potential of the three shells $A$ , $B$ and $C$ . (b) If the shells $A$ and $C$ are at the same potential, obtain the relation between the radii $a$ , $b$ and $c$ .

JEE Advanced 2011

LEVELJEE Advanced

Four point charges, each of $+ q$ , are rigidly fixed at the four corners of a square planar soap film of side $a$ . The surface tension of the soap film is $γ$ . The system of charges and planar film are in equilibrium, and $a = k [\frac{q ^{2}}{γ}]^{1/ N}$ , where $k$ is a constant. Then $N$ is

JEE Advanced 2012

LEVELJEE Advanced

A cubical region of side $a$ has its centre at the origin. It encloses three fixed point charges, $- q$ at $(0, - a /4, 0)$ , $+ 3 q$ at $(0, 0, 0)$ and $- q$ at $(0, + a /4, 0)$ . Choose the correct option(s).

* Multiple Correct Options

(A)

The net electric flux crossing the plane

x = + a /2

is equal to the net electric flux crossing the plane

x = - a /2

(B)

The net electric flux crossing the plane

y = + a /2

is more than the net electric flux crossing the plane

y = - a /2

(C)

The net electric flux crossing the entire region is

q / ε_{0}

(D)

The net electric flux crossing the plane

z = + a /2

is equal to the net electric flux crossing the plane

x = + a /2

Micro-Topic Strategy Matrix

Coulomb's Law

Electric Field

Electric Field Lines, Flux and Gauss's Law

Electric Potential and Potential Difference

Electrostatic Potential Energy

Electric Dipole

Conductors

Capacitance and Capacitors

Combination of Capacitors

Chapter Elite-20 Challenge Zone

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The electric field at r=R is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For a=0, the value of d (maximum value of ρ as shown in the figure) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The electric field within the nucleus is generally observed to be linearly dependent on r. This implies

The electric field at $r = R$ is

For $a = 0$ , the value of $d$ (maximum value of $ρ$ as shown in the figure) is

The electric field within the nucleus is generally observed to be linearly dependent on $r$ . This implies