Electric dipole

Consider an electric dipole comprising two charges $+q$ and $-q$ each with mass $m$, separated by a fixed distance $d$ and initially at rest with its dipole moment pointing along $\hat\imath$. A uniform electric field $E\hat\jmath$ is turned on at time $t=0$ and it is turned off at $t=t_f$, when the dipole moment makes an angle $\theta_f$ with $\hat\imath$. Neglecting any sources of energy loss, correct option(s) is/are:

Two short electric dipoles $A$ and $B$ having dipole moment $p_1$ and $p_2$ respectively are placed with their axis mutually perpendicular as shown in the figure. The resultant electric field at a point $x$ is making an angle of 60° with the line joining points O and $x$. The ratio of the dipole moments $p_2/p_1$ is ________.

A rigid dipole undergoes a simple harmonic motion about its centre in the presence of an electric field $\vec E_1=E_0\hat x$. If another electric field $\vec E_2=2E_0\left(\hat y+\hat z\right)$ is introduced to the system, what will be the percentage change in the frequency of the oscillation (approximate)?

List-I shows four configurations, each consisting of a pair of ideal electric dipoles. Each dipole has a dipole moment of magnitude $p$, oriented as marked by arrows in the figures. In all the configurations the dipoles are fixed such that they are at a distance $2r$ apart along the $x$ direction. The midpoint of the line joining the two dipoles is $X$. The possible resultant electric fields $\vec E$ at $X$ are given in List-II.

Choose the option that describes the correct match between the entries in List-I to those in List-II.

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.]

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?

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.

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 :

Two electric dipoles of dipole moments $1.2\times10^{-30}$ Cm and $2.4\times10^{-30}$ Cm are placed in two different uniform electric fields of strengths $5\times10^4$ NC$^{-1}$ and $15\times10^4$ NC$^{-1}$ respectively. The ratio of maximum torque experienced by the electric dipoles will be $\dfrac1x$. The value of $x$ is ______.

One end of a spring of negligible unstretched length and spring constant $k$ is fixed at the origin (0,0). A point particle of mass $m$ carrying a positive charge $q$ is attached at its other end. The entire system is kept on a smooth horizontal surface. When a point dipole $\vec p$ pointing towards the charge $q$ is fixed at the origin, the spring gets stretched to a length $l$ and attains a new equilibrium position (see figure below). If the point mass is now displaced slightly by $\Delta l \ll l$ from its equilibrium position and released, it is found to oscillate at frequency $\dfrac1\delta\sqrt{\dfrac km}$. The value of $\delta$ is ______.

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)

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)

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.

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 :

Positive and negative point charges of equal magnitude are kept at $\left(0,0,\frac a2\right)$ and $\left(0,0,\frac{-a}2\right)$, respectively. The work done by the electric field when another positive point charge is moved from $(-a,0,0)$ to $(0,a,0)$ is