A point charge $+q$ is placed at the origin. A second point charge $+9 q$ is placed at ($\mathrm{d},0,0$) in Cartesian coordinate system. The point in between them where the electric field vanishes is:

A small bob of mass 100 mg and charge $+10\mu \mathrm{C}$ is connected to an insulating string of length 1 m . It is brought near to an infinitely long non-conducting sheet of charge density ' $\sigma$ ' as shown in figure. If string subtends an angle of $45^{\circ }$ with the sheet at equilibrium the charge density of sheet will be. (Given, ${ϵ}_0=8.85\times 10^{-12}\frac{\mathrm{F}}{\mathrm{m}}$ and acceleration due to gravity, $\mathrm{g}=10\frac{\mathrm{m}}{{\mathrm{s}}^2}$ )

If ${ϵ}_0$ denotes the permittivity of free space and ${\Phi }_E$ is the flux of the electric field through the area bounded by the closed surface, then dimensions of $\left({ϵ}_0\frac{d{\varphi }_E}{dt}\right)$ are that of :

Two charges $q_1$ and $q_2$ are separated by a distance of 30 cm . A third charge $q_3$ initially at ' C ' as shown in the figure, is moved along the circular path of radius 40 cm from C to D . If the difference in potential energy due to movement of $q_3$ from C to D is given by $\frac{q_3\mathrm{K}}{4\pi {ϵ}_0}$, the value of K is :

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Net dipole moment of a polar linear isotropic dielectric substance is not zero even in the absence of an external electric field. Reason (R) : In absence of an external electric field, the different permanent dipoles of a polar dielectric substance are oriented in random directions. In the light of the above statements, choose the most appropriate answer from the options given below :

Two large plane parallel conducting plates are kept 10 cm apart as shown in figure. The potential difference between them is V . The potential difference between the points A and $B$ (shown in the figure) is :

Consider a circular loop that is uniformly charged and has a radius $\mathrm{a}\sqrt{2}$. Find the position along the positive $z$-axis of the cartesian coordinate system where the electric field is maximum if the ring was assumed to be placed in $x y$ plane at the origin :

A dipole with two electric charges of 2 µC magnitude each, with separation distance 0.5 µm, is placed between the plates of a capacitor such that its axis is parallel to an electric field established between the plates when a potential difference of 5 V is applied. Separation between the plates is 0.5 mm. If the dipole is rotated by 30° from the axis, it tends to realign in the direction due to a torque. The value of torque is:

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : The outer body of an aircraft is made of metal which protects persons sitting inside from lightning strikes. Reason (R) : The electric field inside the cavity enclosed by a conductor is zero. In the light of the above statements, choose the most appropriate answer from the options given below :

The electrostatic potential on the surface of uniformly charged spherical shell of radius $\mathrm{R}=10\mathrm{cm}$ is 120 V . The potential at the centre of shell, at a distance $\mathrm{r}=5\mathrm{cm}$ from centre, and at a distance $\mathrm{r}=15$ cm from the centre of the shell respectively, are:

A metallic ring is uniformly charged as shown in figure. AC and BD are two mutually perpendicular diameters. Electric field due to arc $A B$ at ' $O$ ' is ' $E$ ' in magnitude. What would be the magnitude of electric field at ' O ' due to arc ABC ?

Two infinite identical charged sheets and a charged spherical body of charge density ' $\rho$ ' are arranged as shown in figure. Then the correct relation between the electrical fields at $\mathrm{A},\mathrm{B},\mathrm{C}$ and D points is:

Two small spherical balls of mass 10 g each with charges $-2\mu \mathrm{C}$ and $2\mu \mathrm{C}$, are attached to two ends of very light rigid rod of length 20 cm . The arrangement is now placed near an infinite nonconducting charge sheet with uniform charge density of $100\mu \mathrm{C}/{\mathrm{m}}^2$ such that length of rod makes an angle of $30^{\circ }$ with electric field generated by charge sheet. Net torque acting on the rod is: (Take ${\epsilon }_{\mathrm{o}}:8.85\times 10^{-12}{\mathrm{C}}^2/{\mathrm{Nm}}^2$ )

Two projectiles are fired with same initial speed from same point on ground at angles of $(45^{\circ }-\alpha )$ and $(45^{\circ }+\alpha )$, respectively, with the horizontal direction. The ratio of their maximum heights attained is :

A ball of mass 100 g is projected with velocity $20\mathrm{m}/\mathrm{s}$ at $60^{\circ }$ with horizontal. The decrease in kinetic energy of the ball during the motion from point of projection to highest point is

An object of mass ' m ' is projected from origin in a vertical xy plane at an angle $45^{\circ }$ with the $\mathrm{x}-$ axis with an initial velocity ${\mathrm{v}}_0$. The magnitude and direction of the angular momentum of the object with respect to origin, when it reaches at the maximum height, will be [ g is acceleration due to gravity]

The position vector of a moving body at any instant of time is given as $\overrightarrow{\mathrm{r}}=\left(5{\mathrm{t}}^2\hat{i}-5\mathrm{t}\hat{j}\right)\mathrm{m}$. The magnitude and direction of velocity at $t=2 s$ is,

Two balls with same mass and initial velocity, are projected at different angles in such a way that maximum height reached by first ball is 8 times higher than that of the second ball. $T_1$ and $T_2$ are the total flying times of first and second ball, respectively, then the ratio of $T_1$ and $T_2$ is

A river is flowing from west to east direction with speed of $9\mathrm{km}{\mathrm{h}}^{-1}$. If a boat capable of moving at a maximum speed of $27\mathrm{km}{\mathrm{h}}^{-1}$ in still water, crosses the river in half a minute, while moving with maximum speed at an angle of $150^{\circ }$ to direction of river flow, then the width of the river is :

Two projectiles are fired from ground with same initial speeds from same point at angles $(45^{\circ }+$ $\alpha )$ and $\left(45^{\circ }-\alpha \right)$ with horizontal direction. The ratio of their times of flights is