Electric flux and Gauss's law

Two point charges 8 μC and −2 μC are located at $x=2$ cm and $x=4$ cm, respectively on the $x$-axis. The ratio of electric flux due to these charges through two spheres of radii 3 cm and 5 cm with their centers at the origin is ________.

Two co-axial conducting cylinders of same length $\ell$ with radii $\sqrt2R$ 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 dielectric constant $\kappa = 5$. Consider an imaginary plane of the same length $\ell$ 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 ($\epsilon_0$ is the permittivity of free space):

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 oriented at a small angle $\theta$ as shown in the figure. While staying at a distance $r$, the dipole is free to rotate about its center.

If released from rest, then which of the following statement(s) is(are) correct?

[$\varepsilon_0$ is the permittivity of free space.]

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 electric flux through the curved surface becomes $\Phi/\sqrt n$, where the value of $n$ is _____.

An infinitely long thin wire, having a uniform charge density per unit length of $5\,\mathrm{nC/m}$, is passing through a spherical shell of radius $1\,\mathrm{m}$, as shown in the figure. A $10\,\mathrm{nC}$ charge is distributed uniformly over the spherical shell. If the configuration of the charges remains static, the magnitude of the potential difference between points P and R, in Volt, is _____.

[Given: In SI units $\dfrac{1}{4\pi\varepsilon_0} = 9\times10^9$, $\ln 2 = 0.7$. Ignore the area pierced by the wire.]

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 $\dfrac{nq}{6\epsilon_0}$ (in SI units). The value of $n$ is _____.

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 $r_B$, the electrostatic charge density varies as $\rho_B = \dfrac{2k}{r}$. Assume that dimensions are taken care of. All physical quantities are in their SI units.

Which of the following statement(s) is(are) correct?

A long cylindrical volume contains a uniformly distributed charge of density $\rho$. The radius of cylindrical volume is $R$. A charge particle ($q$) revolves around the cylinder in a circular path. The kinetic energy of the particle is :

In the figure, a very large plane sheet of positive charge is shown. $P_1$ and $P_2$ are two points at distance $l$ and $2l$ from the charge distribution. If $\sigma$ is the surface charge density, then the magnitude of electric fields $E_1$ and $E_2$ at $P_1$ and $P_2$ respectively are :

A force of 10 N 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.

If a charge $q$ is placed at the centre of a closed hemispherical non-conducting surface, the total flux passing through the flat surface would be :

The volume charge density of a sphere of radius 6 m is 2 $\mu$C cm$^{-3}$. The number of lines of force per unit surface area coming out from the surface of the sphere is ______ $\times10^{10}$ NC$^{-1}$.

[Given : Permittivity of vacuum $\epsilon_0=8.85\times10^{-12}$ C$^2$ N$^{-1}$ m$^{-2}$]

A long cylindrical volume contains a uniformly distributed charge of density ρ Cm$^{-3}$. The electric field inside the cylindrical volume at a distance $x=\dfrac{2\varepsilon_0}\rho$ m from its axis is ______ Vm$^{-1}$.

A spherically symmetric charge distribution is considered with charge density varying as

$$\rho(r)=\begin{cases}\rho_0\left(\dfrac34-\dfrac rR\right) & \text{for } r\le R\\ \text{zero} & \text{for } r>R\end{cases}$$

Where, r (r < R) is the distance from the centre O (as shown in figure). The electric field at point P will be :

A circular disc of radius $R$ carries surface charge density $\sigma(r) = \sigma_0\left(1 - \dfrac rR\right)$, 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 $\dfrac R4$ and concentric with the disc is $\phi$. Then the ratio $\dfrac{\phi_0}{\phi}$ is ______.

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 equidistant from its top and bottom surfaces. Which of the following option(s) is/are correct?

[$\epsilon_0$ is the permittivity of free space]

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 $O$ of the spherical shell, as shown in the figure. The permittivity of free space is $\epsilon_0$. Which of the following statements is (are) true?

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.

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?

Four closed surfaces and corresponding charge distributions are shown below.

Let the respective electric fluxes through the surfaces be $\Phi_1$, $\Phi_2$, $\Phi_3$ and $\Phi_4$. Then :