Electrostatics — JEE Advanced PYQs

Two charges $Q_1=q$ and $Q_2=mq$ are placed at the points $P_1(a,b)$ and $P_2(ma,mb)$, respectively, in the $XY$ plane, where $a,b\ne 0$ and $m\ne 0,1$. If $V_1$ is the potential at a point in the $XY$ plane due to charge $Q_1$ and …

Consider an electric dipole comprising two charges $+q$ and $-q$ each with mass $m$, separated by a fixed distance $d$ and initially at rest with its dipole moment pointing along $\hat{ı}$. A uniform electric field $E\hat{j}$ is turned on at time $t=0$ and it …

List-I shows four configurations, each consisting of a pair of ideal electric dipoles. Each dipole has a dipole moment of magnitude $p$, oriented as marked by arrows in the figures. In all the configurations the dipoles are fixed such that they are at a distance $2r$ apart a …

Two co-axial conducting cylinders of same length $\ell$ with radii $\sqrt{2}R$ and $2R$ 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 dielec …

A positive point charge of $10^{-8}$ 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 b …

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 ${\sigma }_0$. The separation between any two consecutive sheets is 1 μm. The vario …

A small electric dipole ${\vec{p}}_0$, having a moment of inertia $I$ about its center, is kept at a distance $r$ from the center of a spherical shell of radius $R$. The surface charge density $\sigma$ is uniformly distributed on the spherical shell. The dipole is initially orien …

A charge is kept at the central point P of a cylindrical region. The two edges subtend a half-angle $\theta$ at P, as shown in the figure. When $\theta =30^{\circ }$, then the electric flux through the curved surface of the cylinder is $\Phi$. If $\theta =60^{\circ }$, then the elect …

An infinitely long thin wire, having a uniform charge density per unit length of $5\mathrm{nC}/\mathrm{m}$, is passing through a spherical shell of radius 1 m, as shown in the figure. A 10 nC charge is distributed uniformly over the spherical shell. If the configuration of the …

An electric dipole is formed by two charges $+q$ and $-q$ located in xy-plane at $(0,2)$ mm and $(0,-2)$ mm, respectively, as shown in the figure. The electric potential at point $\mathrm{P}(100,100)$ mm due to the dipole is $V_0$. The charges $+q$ and $-q$ are then moved to th …

Six charges are placed around a regular hexagon of side length a as shown in the figure. Five of them have charge $q$, and the remaining one has charge $x$. The perpendicular from each charge to the nearest hexagon side passes through the center O of the hexagon and is bisect …

A charge $q$ is surrounded by a closed surface consisting of an inverted cone of height $h$ and base radius $R$, and a hemisphere of radius $R$ as shown in the figure. The electric flux through the conical surface is $\frac{nq}{6{ϵ}_0}$ (in SI units). The value …

In the figure, the inner (shaded) region $A$ represents a sphere of radius $r_A=1$, within which the electrostatic charge density varies with the radial distance $r$ from the center as ${\rho }_A=kr$, where $k$ is positive. In the spherical shell $B$ of outer radius …

Two point charges $-Q$ and $+Q/\sqrt{3}$ are placed in the xy-plane at the origin (0,0) and a point (2,0), respectively, as shown in the figure. This results in an equipotential circle of radius $R$ and potential $\mathrm{V}=0$ in the xy-plane with its center at (b, 0). All lengt …

Two point charges $-Q$ and $+Q/\sqrt{3}$ are placed in the xy-plane at the origin (0,0) and a point (2,0), respectively, as shown in the figure. This results in an equipotential circle of radius $R$ and potential $\mathrm{V}=0$ in the xy-plane with its center at (b, 0). All lengt …

A uniform electric field, $\overrightarrow{\mathrm{E}}=-400\sqrt{3}\hat{y}{\mathrm{NC}}^{-1}$ is applied in a region. A charged particle of mass m carrying positive charge q is projected in this region with an initial speed of $2\sqrt{10}\times 10^6{\mathrm{ms}}^{-1}$. This …

One end of a spring of negligible unstretched length and spring constant k is fixed at the origin $(0,0)$. A point particle of mass m carrying a positive charge q is attached at its other end. The entire system is kept on a smooth horizontal surface. When a point dipole …

A circular disc of radius $R$ carries surface charge density $\sigma (r)={\sigma }_0\left(1-\frac{\mathrm{r}}{\mathrm{R}}\right)$, where ${\sigma }_0$ is a constant and $r$ is the distance from the center of the disc. Electric flux through a large spherical surface that encloses the …

Two large circular discs separated by a distance of 0.01 m are connected to a battery via a switch as shown in the figure. Charged oil drops of density $900{\text{ kg m}}^{-3}$ are released through a tiny hole at the center of the top disc. Once some oil drops achieve terminal vel …

A point charge q of mass m is suspended vertically by a string of length $l$. A point dipole of dipole moment $\vec{p}$ 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 an …

Two identical non-conducting solid spheres of same mass and charge are suspended in air from a common point by two non-conducting, massless strings of same length. At equilibrium, the angle between the strings is $\alpha$. The spheres are now immersed in a dielectric liquid of de …

A thin spherical insulating shell of radius R carries a uniformly distributed charge such that the potential at its surface is ${\mathrm{V}}_0$. A hole with a small area $\alpha 4\pi {\mathrm{R}}^2(\alpha \ll 1)$ is made on the shell without affecting the rest of the shell. Whic …

A charged shell of radius R carries a total charge Q. Given $\Phi$ as the flux of electric field through a closed cylindrical surface of height h, radius r 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 …

An electric dipole with dipole moment $\frac{{\mathrm{p}}_0}{\sqrt{2}}(\hat{\mathrm{i}}+\hat{\mathrm{j}})$ is held fixed at the origin O in the presence of an uniform electric field of magnitude ${\mathrm{E}}_0$. If the potential is constant on a circle of radius R centered a …

An infinitely long thin non-conducting wire is parallel to the $z$-axis and carries a uniform line charge density $\lambda$. It pierces a thin non-conducting spherical shell of radius $R$ in such a way that the arc $PQ$ subtends an angle $120^{\circ }$ at the centre …

A particle, of mass $10^{-3}kg$ and charge $1.0C$, is initially at rest. At time $t=0$, the particle comes under the influence of an electric field $\vec{E}(t)=E_0\sin \omega t\hat{ı}$, where $E_0=1.0N{C}^{-1}$ and $\omega =10^{3}rad{s}^{-1}$. Consider the …

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 descr …

A point charge $+Q$ is placed just outside an imaginary hemispherical surface of radius $R$ as shown in the figure. Which of the following statements is/are correct?

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Consider an evacuated cylindrical chamber of height $h$ 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 ar …

An infinitely long uniform line charge distribution of charge per unit length $\lambda$ lies parallel to the $y$-axis in the $y-z$ plane at $z=\frac{\sqrt{3}}{2}a$ (see figure). If the magnitude of the flux of the electric field through the rectangular surface $ABCD$ lying in th …

Consider a uniform spherical charge distribution of radius $R_1$ centred at the origin O. In this distribution, a spherical cavity of radius $R_2$, centred at P with distance $OP=a=R_1-R_2$ (see figure) is made. If the electric field inside the cavity at position $\vec{r}$ …

Charges $Q,2Q$ and $4Q$ are uniformly distributed in three dielectric solid spheres 1, 2 and 3 of radii $R/2,R$ and $2R$, respectively, as shown in figure. If magnitudes of the electric fields at point P at a distance $R$ from the center of spheres 1, 2 and 3 are …

Four charges $Q_1,Q_2,Q_3$ 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 g …

Let $E_1(r),E_2(r)$ and $E_3(r)$ be the respective electric fields at a distance $r$ from a point charge $Q$, an infinitely long wire with constant linear charge density $\lambda$, and an infinite plane with uniform surface charge density $\sigma$. If …

Two non-conducting solid spheres of radii $R$ and $2R$, having uniform volume charge densities ${\rho }_1$ and ${\rho }_2$ respectively, touch each other. The net electric field at a distance $2R$ from the centre of the smaller sphere, along the line joining the centres …

Two non-conducting spheres of radii $R_1$ and $R_2$ and carrying uniform volume charge densities $+\rho$ and $-\rho$, respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region,

Consider a thin spherical shell of radius $R$ with its centre at the origin, carrying uniform positive surface charge density. The variation of the magnitude of the electric field $|\vec{E}(r)|$ and the electric potential $V(r)$ with the distance $r$ from the centre, is b …

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

An infinitely long solid cylinder of radius $R$ has a uniform volume charge density $\rho$. It has a spherical cavity of radius $R/2$ with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the electric field at the point $P$, which is at a …

Six point charges are kept at the vertices of a regular hexagon of side $L$ and centre $O$, as shown in the figure. Given that $K=\frac{1}{4\pi {\epsilon }_0}\frac{q}{L^2}$, which of the following statement(s) is(are) correct ?

Consider an electric field $\vec{E}=E_0\hat{x}$, where $E_0$ is a constant. The flux through the shaded area (as shown in the figure) due to this field is

A spherical metal shell A of radius ${\mathrm{R}}_{\mathrm{A}}$ and a solid metal sphere B of radius ${\mathrm{R}}_{\mathrm{B}}\left(<{\mathrm{R}}_{\mathrm{A}}\right)$ are kept far apart and each is given charge '+Q'. Now they are connected by a thin metal wire. Then

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 $\gamma$. The system of charges and planar film are in equilibrium, and $a=k{\left[\frac{q^2}{\gamma }\right]}^{1/N}$, …

Which of the field patterns given below is valid for electric field as well as for magnetic field?

Which of the following statement(s) is/are correct?
(A) If the electric field due to a point charge varies as $r^{-25}$ instead of $r^{-2}$, then the Gauss law will still be valid.
(B) The Gauss law can be used to calculate the field distribution around an electric dipole.
(C …

A uniformly charged thin spherical shell of radius $R$ carries uniform surface charge density of $\sigma$ per unit area. It is made of two hemispherical shells, held together by pressing them with force $F$ (see figure). $F$ is proportional to

A solid sphere of radius $R$ has a charge $Q$ distributed in its volume with a charge density $\rho =\kappa r^a$, where $\kappa$ and $a$ are constants and $r$ is the distance from its centre. If the electric field at $r=\frac{R}{2}$ is $\frac{1}{8}$ times that at …

Three concentric metallic spherical shells of radii $R,2R,3R$, are given charges $Q_1,Q_2,Q_3$, respectively. It is found that the surface charge densities on the outer surfaces of the shells are equal. Then, the ratio of the charges given to the shells, …

Six point charges, each of the same magnitude $q$, are arranged in different manners as shown in Column II. In each case, a point M and a line PQ passing through M are shown. Let $E$ be the electric field and $V$ be the electric potential at M (potential at infinity is zero …

Consider a system of three charges $\frac{q}{3}$, $\frac{q}{3}$ and $-\frac{2q}{3}$ placed at points A, B and C, respectively, as shown in the figure. Take O to be the centre of the circle of radius $R$ and angle $\mathrm{CAB}=60^{\circ }$