Electric field

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)

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 $\pi=3.14$)

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.

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)

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

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 :

Two point charges Q each are placed at a distance d apart. A third point charge q is placed at a distance $x$ from mid-point on the perpendicular bisector. The value of $x$ at which charge q will experience the maximum Coulomb's force is:

Given below are two statements :

Statement I: The electric force changes the speed of the charged particle and hence changes its kinetic energy; whereas the magnetic force does not change the kinetic energy of the charged particle.

Statement II: The electric force accelerates the positively charged particle perpendicular to the direction of electric field. The magnetic force accelerates the moving charged particle along the direction of magnetic field.

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

A charge q is uniformly distributed along the length L of a rod. It is then bent in the shape of a semicircle. The magnitude of electric field at the centre of semicircle will be :

A cube of side 'a' has point charges +Q located at each of its vertices except at the origin where the charge is −Q. The electric field at the centre of cube is :

A uniform electric field, $\vec E = -400\sqrt3\,\hat y\ \mathrm{NC^{-1}}$ is applied in a region. A charged particle of mass $m$ carrying positive charge $q$ is projected in this region with an initial speed of $2\sqrt{10}\times10^{6}\ \mathrm{m\,s^{-1}}$. This particle is aimed to hit a target T, which is 5 m away from its entry point into the field as shown schematically in the figure. Take $\dfrac qm = 10^{10}\ \mathrm{C\,kg^{-1}}$. Then

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 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?

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?