Electric flux and Gauss's law

An infinite line charge of uniform electric charge density $\lambda$ lies along the axis of an electrically conducting infinite cylindrical shell of radius $R$. At time $t=0$, the space inside the cylinder is filled with a material of permittivity $\varepsilon$ and electrical conductivity $\sigma$. 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 $j(t)$ at any point in the material?

The region between two concentric spheres of radii '$a$' and '$b$', respectively (see figure), has volume charge density $\rho=\dfrac Ar$, where A is a constant and r is the distance from the centre. At the centre of the spheres is a point charge Q. The value of A such that the electric field in the region between the spheres will be constant, is :

The potential (in volts) of a charge distribution is given by

$V(z)=30-5z^2$ for |z| ≤ 1 m

V(z) = 35 − 10 |z| for |z| ≥ 1 m.

V(z) does not depend on x and y. If this potential is generated by a constant charge per unit volume $\rho_0$ (in units of $\epsilon_0$) which is spread over a certain region, then choose the correct statement.

Within a spherical charge distribution of charge density ρ(r), N equipotential surfaces of potential $V_0$, $V_0+\Delta V$, $V_0+2\Delta V$, .......... $V_0+N\Delta V$ (ΔV > 0), are drawn and have increasing radii $r_0$, $r_1$, $r_2$, .......... $r_N$, respectively. If the difference in the radii of the surfaces is constant for all values of $V_0$ and ΔV then :

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 $z=\dfrac{\sqrt3}{2}a$ (see figure). If the magnitude of the flux of the electric field through the rectangular surface $ABCD$ lying in the $x$-$y$ plane with its centre at the origin is $\dfrac{\lambda L}{n\varepsilon_0}$ ($\varepsilon_0$ = permittivity of free space), then the value of $n$ is

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 $\mathrm{OP}=a=R_1-R_2$ (see figure) is made. If the electric field inside the cavity at position $\vec r$ is $\vec E(\vec r)$, then the correct statement(s) is(are)

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

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 $E_1$, $E_2$ and $E_3$ respectively, then

The magnitude of the average electric field normally present in the atmosphere just above the surface of the Earth is about 150 N/C, directed inward towards the center of the Earth. This gives the total net surface charge carried by the Earth to be :

[Given $\epsilon_o=8.85\times10^{-12}$ C$^2$/N-m$^2$, $R_E=6.37\times10^6$ m]

A cone of base radius R and height $h$ is located in a uniform electric field $\vec E$ parallel to its base. The electric flux entering the cone is :

A spherically symmetric charge distribution is characterised by a charge density having the following variation :
$$\rho(r)=\rho_o\left(1-\frac rR\right)\ \text{for } r<R,\qquad\rho(r)=0\ \text{for } r\ge R$$
Where r is the distance from the centre of the charge distribution and $\rho_o$ is a constant. The electric field at an internal point (r < R) is :

The electric field in a region of space is given by, $\vec E=E_o\hat i+2E_o\hat j$ where $E_0=100$ N/C. The flux of this field through a circular surface of radius 0.02 m parallel to the Y-Z plane is nearly :

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 of the spheres, is zero. The ratio $\dfrac{\rho_1}{\rho_2}$ can be

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 best represented by which graph?

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 distance $2R$ from the axis of the cylinder, is given by the expression $\dfrac{23\rho R}{16k\varepsilon_0}$. The value of $k$ is

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

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

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, $Q_1:Q_2:Q_3$, is