Electrostatics

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 :

If the electric potential at any point $(x,y,z)$ m in space is given by $V=3x^2$ volt. The electric field at the point $(1,0,3)$ m will be :

Eight similar drops of mercury are maintained at 12 V each. All these spherical drops combine into a single big drop. The potential energy of bigger drop will be ______ E. Where E is the potential energy of a single smaller drop.

The volume charge density of a sphere of radius 6 m is 2 $\mu$C cm$^{-3}$. The number of lines of force per unit surface area coming out from the surface of the sphere is ______ $\times10^{10}$ NC$^{-1}$.

[Given : Permittivity of vacuum $\epsilon_0=8.85\times10^{-12}$ C$^2$ N$^{-1}$ m$^{-2}$]

Two uniformly charged spherical conductors A and B of radii 5 mm and 10 mm are separated by a distance of 2 cm. If the spheres are connected by a conducting wire, then in equilibrium condition, the ratio of the magnitudes of the electric fields at the surface of the sphere A and B will be :

Three point charges of magnitude 5 μC, 0.16 μC and 0.3 μC are located at the vertices A, B, C of a right angled triangle whose sides are AB = 3 cm, BC = $3\sqrt2$ cm and CA = 3 cm and point A is the right angle corner. Charge at point A experiences ______ N of electrostatic force due to the other two charges.

Two identical positive charges Q each are fixed at a distance of '2a' apart from each other. Another point charge $q_0$ with mass 'm' is placed at midpoint between two fixed charges. For a small displacement along the line joining the fixed charges, the charge $q_0$ executes SHM. The time period of oscillation of charge $q_0$ will be :

A long cylindrical volume contains a uniformly distributed charge of density ρ Cm$^{-3}$. The electric field inside the cylindrical volume at a distance $x=\dfrac{2\varepsilon_0}\rho$ m from its axis is ______ Vm$^{-1}$.

A charge of 4 μC is to be divided into two. The distance between the two divided charges is constant. The magnitude of the divided charges so that the force between them is maximum, will be :

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 ______.

A spherically symmetric charge distribution is considered with charge density varying as

$$\rho(r)=\begin{cases}\rho_0\left(\dfrac34-\dfrac rR\right) & \text{for } r\le R\\ \text{zero} & \text{for } r>R\end{cases}$$

Where, r (r < R) is the distance from the centre O (as shown in figure). The electric field at point P will be :

Given below are two statements.

Statement I : Electric potential is constant within and at the surface of each conductor.

Statement II : Electric field just outside a charged conductor is perpendicular to the surface of the conductor at every point.

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

Two identical metallic spheres A and B when placed at certain distance in air repel each other with a force of F. Another identical uncharged sphere C is first placed in contact with A and then in contact with B and finally placed at midpoint between spheres A and B. The force experienced by sphere C will be:

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 :

The value of $R$ is ____ meter.

The value of $b$ is ____ meter.

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

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 ______.