Let a total charge $2Q$ be distributed in a sphere of radius $R$, with the charge density given by $\rho (r)=kr$, 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 form the centre. If $A$ and $B$ do not experience any force, then:

Two infinite planes each with uniform surface charge density $+\sigma$ are kept in such a way that the angle between them is $30^0$. The electric field in the region shown between them is given by :

A particle of mass $m$ and charge $q$ is released from rest in a uniform electric field. If there is no other force on the particle, the dependence of its speed $v$ on the distance $x$ travelled by it is correctly given by (graphs are schematic and not drawn to scale)

In finding the electric field using Gauss law the formula $|\vec{E}|=\frac{q_{enc}}{{\in }_0|A|}$ is applicable. In the formula ${\in }_0$ is permittivity of free space, $A$ is the area of Gaussian surface and $q_{enc}$ is charge enclosed by the Gaussian surface. This equation can be used in which of the following situation?

Three charged particles $A,B$ and $C$ with charges $-4q,2q$ and $-2q$ are present on the circumference of a circle of radius $d$. The charged particles $A,C$ and centre $O$ of the circle formed an equilateral triangle as shown in figure. Electric field at $O$ along x-direction is:

An electric dipole of moment $\vec{p}=(-\hat{i}-3\hat{j}+2\hat{k})\times 10^{-29}$ C.m is at the origin $(0,0,0)$. The electric field due to this dipole at $\vec{r}=+\hat{i}+3\hat{j}+5k$ (note that $\vec{r}\cdot \vec{p}=0$ is parallel to):

An electric field $\vec{E}=4x\hat{i}-(y^2+1)\hat{j}N/C$ passes through the box shown in figure. The flux of the electric field through surfaces $ABCD$ and $BCGF$ are marked as ${\varphi }_I$ and ${\varphi }_{II}$ respectively. The difference between $({\varphi }_I-{\varphi }_{II})$ is (in $N{m}^2/C)$

A body of mass $M$ and charge $q$ is connected to a spring of spring constant $k$. It is oscillating along x-direction about its equilibrium position, taken to be at $x=0$, with an amplitude $A$. An electric field $E$ is applied along the x-direction. Which of the following statements is correct?

A charge $Q$ is placed at a distance $a/2$ above the centre of the square surface of edge $a$ as shown in the figure.
The electric flux through the square surface is:

Four closed surface and corresponding charges distributions are shown below:
Let the respective electric fluxes through the surfaces be ${\Phi }_1$, ${\Phi }_2$, ${\Phi }_3$ and ${\Phi }_4$. Then :

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 E${}_1(r_0)=E_2(r_0)=E_3(r_0)$ at a given distance r${}_0$, then :

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? 

A particle of mass $10^{-3}$ kg and charge $1.0$ C, is initially at rest. At time $t=0$, the particle comes under the influence an electric field $\vec{E}(t)=E_{0}sin(\omega t)\hat{i}$ where $E_0=1.0N{C}^{-1}$ and $\omega =10^{3}rad{s}^{-1}$. Consider the effect of only the electrical force on the particle. Then the maximum speed, in $m{s}^{-1}$, attained by the particle at subsequent times is __________.

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	List - II
(P) E is independent of d	(1) A point charge Q at the origin
(Q) $E\propto \frac{1}{d}$	(2) A small dipole with point charges Q at $(0,0,l)$ and - Q at (0, 0, -l). Take $2l<<d$
(R) $E\propto \frac{1}{d^2}$	(3) A infinite line charge coincident with the x-axis, with uniform linear charge density $\lambda$.
(S) $E\propto \frac{1}{d^3}$	(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 + $\lambda$ and the one along $(y=0,z=-l)$ has a charge density $-\lambda$. Take $2l<<d$
	(5) Infinite plane charge coincident with the xy-plane with uniform surface charge density.

A charged shell of radius R carries a total charge Q. Given $\varphi$ as the flux of electric field through a closed cylindrical surface of height h, radius r and with its centre same as that of the shell. Here center of 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? $[{\epsilon }_0$ is the permittivity of free space$]$

A uniform electric field, $\vec{E}=-400\sqrt{3}yN{C}^{-1}$ 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 $2\sqrt{10}\times 10^{6}m{s}^{-1}$. This particle is aimed to hit a target T, which is 5 m away from its entry point into the field as shown schematically in the figure.
Take $\frac{q}{m}=10^{10}Ck{g}^{-1}$. Then

A circular disc of radius R carries surface charge density $\sigma (r)={\sigma }_0\left(1-\frac{r}{R}\right)$ 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 charged disc completely is ${\Phi }_0$. Electric flux through another spherical surface of radius $R/4$ and concentric with the disc is $\Phi$. Then the ratio $\frac{{\Phi }_0}{\Phi }$ is________.

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 center 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 distance 2R from the axis of the cylinder, is given by the expression $\frac{23\rho R}{16k{ϵ}_0}$. The value of k is:

Two charges, each equal to $q$, are kept at $x=-a$ and $x=a$ on the $x$-axis. A particle of mass $m$ and charge $q_0=\frac{q}{2}$ is placed at the origin. If charge $q_0$ is given a small displacement $(y<<a)$ along the $y$-axis, the net force acting on the particle is proportional to: