Electrostatics

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.]

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 P $(100,100)$ mm due to the dipole is $V_0$. The charges $+q$ and $-q$ are then moved to the points $(-1,2)$ mm and $(1,-2)$ mm, respectively. What is the value of electric potential at P due to the new dipole?

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 bisected by the side.

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

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 disk of radius $R$ with uniform positive charge density $\sigma$ is placed on the $xy$ plane with its center at the origin. The Coulomb potential along the $z$-axis is
$$V(z) = \frac{\sigma}{2\epsilon_0}\left(\sqrt{R^2 + z^2} - z\right).$$
A particle of positive charge $q$ is placed initially at rest at a point on the $z$ axis with $z = z_0$ and $z_0 > 0$. In addition to the Coulomb force, the particle experiences a vertical force $\vec F = -c\,\hat k$ with $c > 0$. Let $\beta = \dfrac{2c\epsilon_0}{q\sigma}$. Which of the following statement(s) is(are) correct?

A vertical electric field of magnitude $4.9\times10^5$ N/C just prevents a water droplet of a mass 0.1 g from falling. The value of charge on the droplet will be :

(Given $g=9.8\ \mathrm{m/s^2}$)

Two identical charged particles each having a mass 10 g and charge $2.0\times10^{-7}$ C are placed on a horizontal table with a separation of L between them such that they stay in limited equilibrium. If the coefficient of friction between each particle and the table is 0.25, find the value of L. [Use $g=10\ \mathrm{ms^{-2}}$]

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 :

27 identical drops are charged at 22 V each. They combine to form a bigger drop. The potential of the bigger drop will be ______ V.

Given below two statements : One is labelled as Assertion (A) and other is labelled as Reason (R).

Assertion (A) : Non-polar materials do not have any permanent dipole moment.

Reason (R) : When a non-polar material is placed in an electric field, the centre of the positive charge distribution of it's individual atom or molecule coincides with the centre of the negative charge distribution.

In the light of above statements, choose the most appropriate answer from the options given below.

Sixty four conducting drops each of radius 0.02 m and each carrying a charge of $5\ \mu$C are combined to form a bigger drop. The ratio of surface density of bigger drop to the smaller drop will be :

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 :

Three identical charged balls each of charge 2 C are suspended from a common point P by silk threads of 2 m each (as shown in figure). They form an equilateral triangle of side 1m.

The ratio of net force on a charged ball to the force between any two charged balls will be :

Given below are two statements :

Statement-I : A point charge is brought in an electric field. The value of electric field at a point near to the charge may increase if the charge is positive.

Statement-II : An electric dipole is placed in a non-uniform electric field. The net electric force on the dipole will not be zero.

Choose the correct answer from the options given below :

The three charges $q/2$, $q$ and $q/2$ are placed at the corners A, B and C of a square of side '$a$' as shown in figure. The magnitude of electric field (E) at the corner D of the square, is :

Two point charges A and B of magnitude $+8\times10^{-6}$ C and $-8\times10^{-6}$ C respectively are placed at a distance d apart. The electric field at the middle point O between the charges is $6.4\times10^4$ NC$^{-1}$. The distance 'd' between the point charges A and B is :

A positive charge particle of 100 mg is thrown in opposite direction to a uniform electric field of strength $1\times10^5$ NC$^{-1}$. If the charge on the particle is 40 $\mu$C and the initial velocity is 200 ms$^{-1}$, how much distance it will travel before coming to the rest momentarily :