Electrostatics

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 ______.

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 $900\ \mathrm{kg\,m^{-3}}$ are released through a tiny hole at the center of the top disc. Once some oil drops achieve terminal velocity, the switch is closed to apply a voltage of 200 V across the discs. As a result, an oil drop of radius $8\times10^{-7}$ m stops moving vertically and floats between the discs. The number of electrons present in this oil drop is ______. (neglect the buoyancy force, take acceleration due to gravity $= 10\ \mathrm{m\,s^{-2}}$ and charge on an electron $(e) = 1.6\times10^{-19}$ C)

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 angles and distances is shown in the figure below. If the work done in bringing the dipole to this position is $N\times(mgh)$, 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, $\dfrac{F}{\sin\theta}$ is the same for all forces, where $F$ is any one of the forces and $\theta$ is the angle between the other two forces)

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 density $800\ \mathrm{kg\,m^{-3}}$ and dielectric constant 21. If the angle between the strings remains the same after the immersion, then

A thin spherical insulating shell of radius $R$ carries a uniformly distributed charge such that the potential at its surface is $V_0$. A hole with a small area $\alpha4\pi R^2\ (\alpha \ll 1)$ is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?

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 electric dipole with dipole moment $\dfrac{p_0}{\sqrt2}(\hat i + \hat j)$ is held fixed at the origin O in the presence of an uniform electric field of magnitude $E_0$. If the potential is constant on a circle of radius $R$ centered at the origin as shown in figure, then the correct statement(s) is/are:

($\epsilon_0$ is permittivity of free space. $R \gg$ dipole size)

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?

An electric dipole has a fixed dipole moment $\vec p$, which makes angle θ with respect to $x$-axis. When subjected to an electric field $\vec E_1=E\hat i$, it experiences a torque $\vec T_1=\tau\hat k$. When subjected to another electric field $\vec E_2=\sqrt3E_1\hat j$ it experiences a torque $\vec T_2=-\vec T_1$. The angle θ is :

There is a uniform electrostatic field in a region. The potential at various points on a small sphere centred at P, in the region, is found to vary between the limits 589.0 V to 589.8 V. What is the potential at a point on the sphere whose radius vector makes an angle of 60° with the direction of the field ?

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 :

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 electrostatic energy of $Z$ protons uniformly distributed throughout a spherical nucleus of radius $R$ is given by
$$E=\frac35\,\frac{Z(Z-1)e^2}{4\pi\varepsilon_0R}$$
The measured masses of the neutron, $^1_1\mathrm H$, $^{15}_{7}\mathrm N$ and $^{15}_{8}\mathrm O$ are 1.008665 u, 1.007825 u, 15.000109 u and 15.003065 u, respectively. Given that the radii of both the $^{15}_{7}\mathrm N$ and $^{15}_{8}\mathrm O$ nuclei are same, $1\ \mathrm u=931.5\ \mathrm{MeV}/c^2$ ($c$ is the speed of light) and $e^2/(4\pi\varepsilon_0)=1.44$ MeV fm. Assuming that the difference between the binding energies of $^{15}_{7}\mathrm N$ and $^{15}_{8}\mathrm O$ is purely due to the electrostatic energy, the radius of either of the nuclei is

($1\ \mathrm{fm}=10^{-15}$ m)

Which one of the following statements is correct?

The average current in the steady state registered by the ammeter in the circuit will be

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 :

A combination of capacitors is set up as shown in the figure. The magnitude of the electric field, due to a point charge Q (having a charge equal to the sum of the charges on the 4 μF and 9 μF capacitors), at a point distant 30 m from it, would equal :

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.