Electrostatics

Two charges $Q_1=q$ and $Q_2=mq$ are placed at the points $P_1(a,b)$ and $P_2(ma,mb)$, respectively, in the $XY$ plane, where $a,b\ne0$ and $m\ne0,1$. If $V_1$ is the potential at a point in the $XY$ plane due to charge $Q_1$ and $V_2$ is the potential at that point due to charge $Q_2$. Correct statement(s) for the points at which $|V_1|=|V_2|$ is/are:

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 charged conducting spheres $S_1$ and $S_2$ of radii 8 cm and 18 cm are connected to each other by a wire. After equilibrium is established, the ratio of electric fields on $S_1$ and $S_2$ spheres are $E_{S1}$ and $E_{S2}$ respectively. The value of $\dfrac{E_{S1}}{E_{S2}}$ is ________.

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 ________.

Two metal plates ($A$, $B$) are kept horizontally with separation of $\left(\dfrac{12}\pi\right)$ cm, with plate $A$ on the top. An atomizer jet sprays oil (density 1.5 g/cm³) droplets of radius 1 mm horizontally. All oil droplets carry a charge 5 nC. The potentials $V_A$ and $V_B$ are required on plates A and B respectively in order to ensure the droplets do not descend. The values of $V_A$ and $V_B$ are ________.
(Neglect the air resistance to the droplets and take $g=10$ m/s²)

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 ________.

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)?

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R

Assertion A: In electrostatics, a conductor does not store any net charge inside.

Reason R: Inside the capacitor (with no dielectric medium), the free charge carriers, if placed between the plates of capacitor, experience force and drift.

Choose the correct answer from the options given below

An electron of mass $m$ is moving in an electric field $\vec E=-2E_0\hat i$ ($E_0$ = constant > 0), with an initial velocity $\vec V=v_0\hat i$ ($v_0$ = constant > 0). If $\lambda_0=\dfrac h{4mv_0}$, its de Broglie wavelength at time $t$ is ________.
($e$ = charge of electron)

A thin half ring of radius 35 cm is uniformly charged with a total charge of $Q$ coulomb. If the magnitude of the electric field at centre of the half ring is 100 V/m, then the value of $Q$ is ________ nC.

($\epsilon_o=8.85\times10^{-12}$ C²/Nm² and $\pi=3.14$)

A three coulomb charge moves from the point $(0,-2,-5)$ to the point $(5,1,2)$ in an electric field expressed as $\vec E=2x\hat i+3y^2\hat j+4\hat k$ N/C. The work done in moving the charge is ________ J.

The electric potential as a function of $x$, $y$ is given by $V=5(x^2-y^2)$ V. The electric field at a point $(2,3)$ m is ________ V/m.

Two point charges $q_1=3\ \mu$C and $q_2=-4\ \mu$C are placed at points $\left(2\hat i+3\hat j+3\hat k\right)$ and $\left(\hat i+\hat j+\hat k\right)$ respectively. Force on charge $q_2$ is ________ N. $\left(\text{Take }\dfrac1{4\pi\epsilon_0}=9\times10^9\text{ SI Units}\right)$

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.

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 positive point charge of $10^{-8}$ C is kept at a distance of 20 cm from the center of a neutral conducting sphere of radius 10 cm. The sphere is then grounded and the charge on the sphere is measured. The grounding is then removed and subsequently the point charge is moved by a distance of 10 cm further away from the center of the sphere along the radial direction. Taking $\dfrac{1}{4\pi\epsilon_0} = 9\times10^9\ \mathrm{Nm^2/C^2}$ (where $\epsilon_0$ is the permittivity of free space), which of the following statements is/are correct:

Six infinitely large and thin non-conducting sheets are fixed in configurations I and II. As shown in the figure, the sheets carry uniform surface charge densities which are indicated in terms of $\sigma_0$. The separation between any two consecutive sheets is 1 $\mu$m. The various regions between the sheets are denoted as 1, 2, 3, 4 and 5. If $\sigma_0 = 9\ \mu\mathrm{C/m^2}$, then which of the following statements is/are correct:

(Take permittivity of free space $\epsilon_0 = 9\times10^{-12}$ F/m)

Two beads, each with charge $q$ and mass $m$, are on a horizontal, frictionless, non-conducting, circular hoop of radius $R$. One of the beads is glued to the hoop at some point, while the other one performs small oscillations about its equilibrium position along the hoop. The square of the angular frequency of the small oscillations is given by

[$\varepsilon_0$ 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 _____.