Two point charges of 1 nC and 2 nC are placed at the two corners of equilateral triangle of side 3 cm . The work done in bringing a charge of 3 nC from infinity to the third corner of the triangle is $\_\_\_\_$ $\mu \mathrm{J}$.

$ \frac{1}{4 \pi \epsilon_0}=9 \times 10^9 \mathrm{~N} \cdot \mathrm{~m}^2 / \mathrm{C}^2 $

The electrostatic potential in a charged spherical region of radius $r$ varies as $V=a r^3+b$, where $a$ and $b$ are constants. The total charge in the sphere of unit radius is $\alpha \times \pi a \in_0$. The value of $\alpha$ is $\_\_\_\_$ .

(permittivity of vacuum is $\epsilon_0$ )

Three charges $+2 q,+3 q$ and $-4 q$ are situated at $(0,-3 a),(2 a, 0)$ and $(-2 a, 0)$ respectively in the $x y$ plane. The resultant dipole moment about origin is $\_\_\_\_$ .

Two shorts dipoles $(A, B), A$ having charges $\pm 2 \mu \mathrm{C}$ and length 1 cm and $B$ having charges $\pm 4 \mu \mathrm{C}$ and length 1 cm are placed with their centres 80 cm apart as shown in the figure. The electric field at a point $P$, equi-distant from the centres of both dipoles is $\_\_\_\_$ N/C.

Two charges $7 \mu \mathrm{C}$ and $-2 \mu \mathrm{C}$ are placed at $(-9,0,0) \mathrm{cm}$ and $(9,0,0) \mathrm{cm}$ respectively in an external field $E=\frac{\mathrm{A}}{r^2} \hat{r}$, where $A=9 \times 10^5 \mathrm{~N} / \mathrm{C} . \mathrm{m}^2$. Considering the potential at infinity is 0 , the electrostatic energy of the configuration is $\_\_\_\_$ J.

Two point charges $2 q$ and $q$ are placed at vertex $A$ and centre of face $C D E F$ of the cube as shown in figure. The electric flux passing through the cube is:

Five positive charges each having charge $q$ are placed at the vertices of a pentagon as shown in the figure. The electric potential $(V)$ and the electric field $(\vec{E})$ at the center $O$ of the pentagon due to these five positive charges are :

Three small identical bubbles of water having same charge on each coalesce to form a bigger bubble. Then the ratio of the potentials on one initial bubble and that on the resultant bigger bubble is :

Electric field in a region is given by $\vec{E}=A x \hat{i}+B y \hat{j}$, where $A=10 \mathrm{~V} / \mathrm{m}^2$ and $B=5 \mathrm{~V} / \mathrm{m}^2$. If the electric potential at a point $(10,20)$ is 500 V , then the electric potential at origin is $\_\_\_\_$ V.

A simple pendulum has a bob with mass $m$ and charge $q$. The pendulum string has negligible mass. When a uniform and horizontal electric field $\vec{E}$ is applied, the tension in the string changes. The final tension in the string, when pendulum attains an equilibrium position is $\_\_\_\_$ .

Six point charges are kept $60^{\circ}$ apart from each other on the circumference of a circle of radius $R$ as shown in figure. The net electric field at the center of the circle is $\_\_\_\_$ .

( $\epsilon_0$ is permittivity of free space)

Consider two identical metallic spheres of radius $R$ each having charge $Q$ and mass $m$. Their centers have an initial separation of $4R$. Both the spheres are given an initial speed of $u$ towards each other. The minimum value of $u$, so that they can just touch each other is:

(Take $k = \frac{1}{4 \pi \epsilon_0}$ and assume $kQ^2 > Gm^2$ where $G$ is the Gravitational constant)

A point charge of $10^{-8} \mathrm{C}$ is placed at origin. The work done in moving a point charge $2 \mu \mathrm{C}$ from point $A(4,4,2) \mathrm{m}$ to $B(2,2,1) \mathrm{m}$ is $\_\_\_\_$ J. $\left(\frac{1}{4 \pi \epsilon_{\mathrm{o}}}=9 \times 10^9\right.$ in SI units)

Two point charges $\mathrm{q}_1=3 \mu C$ and $\mathrm{q}_2=-4 \mu C$ are placed at points $(2 \hat{i}+3 \hat{j}+3 \hat{k})$ and $(\hat{i}+\hat{j}+\hat{k})$ respectively. Force on charge $\mathrm{q}_2$ is $\_\_\_\_$ N. (Take $\frac{1}{4 \pi \epsilon_0}=9 \times 10^9$ SI Units)

The electric potential as a function of $x, y$ is given by $V=5\left(x^2-y^2\right) V$. The electric field at a point $(2,3) \mathrm{m}$ is $\_\_\_\_$ $\mathrm{V} / \mathrm{m}$.

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 $100 \mathrm{~V} / \mathrm{m}$, then the value of $Q$ is $\_\_\_\_$ nC .

$ \left(\epsilon_0=8.85 \times 10^{-12} \mathrm{C}^2 / \mathrm{Nm}^2 \text { and } \pi=3.14\right) $

A rigid dipole undergoes a simple harmonic motion about its centre in the presence of an electric field $\overrightarrow{\mathrm{E}}_1=\mathrm{E}_0 \hat{x}$. If another electric field $\overrightarrow{\mathrm{E}}_2=2 \mathrm{E}_0(\hat{y}+\hat{z})$ 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 $\mathbf{R}$

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

Two metal plates (A, B) are kept horizontally with separation of $ \left( \frac{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 $8\ \mu C$ and $-2\ \mu C$ 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 ________.

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 $60^{\circ}$ with the line joining points $O$ and $x$. The ratio of the dipole moments $p_2/p_1$ is _________.

Two metal spheres of radius R and 3R have same surface charge density σ. If they are brought in contact and then separated, the surface charge density on smaller and bigger sphere becomes σ₁ and σ₂, respectively. The ratio $ \frac{\sigma_1}{\sigma_2} $ is

Given below are two statements: one is labelled as Assertion $\mathbf{A}$ and the other is labelled as Reason $\mathbf{R}$

Assertion A : Work done in moving a test charge between two points inside a uniformly charged spherical shell is zero, no matter which path is chosen.

Reason R : Electrostatic potential inside a uniformly charged spherical shell is constant and is same as that on the surface of the shell.

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

Electric charge is transferred to an irregular metallic disk as shown in the figure. If $\sigma_1$, $\sigma_2$, $\sigma_3$ and $\sigma_4$ are charge densities at given points then, choose the correct answer from the options given below:

A. $\sigma_1>\sigma_3 ; \sigma_2=\sigma_4$

B. $\sigma_1>\sigma_2 ; \sigma_3>\sigma_4$

C. $\sigma_1>\sigma_3>\sigma_2=\sigma_4$

D. $\sigma_1<\sigma_3<\sigma_2=\sigma_4$

E. $\sigma_1=\sigma_2=\sigma_3=\sigma_4$

An infinitely long wire has uniform linear charge density $\lambda = 2 \text{ nC/m}$. The net flux through a Gaussian cube of side length $\sqrt{3}$ cm, if the wire passes through any two corners of the cube, that are maximally displaced from each other, would be $x \text{ Nm}^2\text{C}^{-1}$, where $x$ is:

[Neglect any edge effects and use $\frac{1}{4\pi \epsilon_0} = 9 \times 10^9$ SI units]

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 :

If $\epsilon_0$ 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 $\left(\epsilon_0 \frac{d \phi_E}{d t}\right)$ 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 $\frac{q_3 \mathrm{~K}}{4 \pi \epsilon_0}$, the value of K is :

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 $\varepsilon_{\mathrm{o}}: 8.85 \times 10^{-12} \mathrm{C}^2 / \mathrm{Nm}^2$ )

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:

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 :

Consider two infinitely large plane parallel conducting plates as shown below. The plates are uniformly charged with a surface charge density $+\sigma$ and $-2 \sigma$. The force experienced by a point charge +q placed at the mid point between two plates will be:

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, $\epsilon_0=8.85 \times 10^{-12} \frac{\mathrm{~F}}{\mathrm{~m}}$ and acceleration due to gravity, $\mathrm{g}=10 \frac{\mathrm{~m}}{\mathrm{~s}^2}$ )

A point charge causes an electric flux of $-2 \times 10^4 \mathrm{Nm}^2 \mathrm{C}^{-1}$ to pass through a spherical Gaussian surface of 8.0 cm radius, centred on the charge. The value of the point charge is :

(Given $\epsilon_0=8.85 \times 10^{-12} \mathrm{C}^2 \mathrm{~N}^{-1} \mathrm{~m}^{-2}$ )

An electric dipole is placed at a distance of 2 cm from an infinite plane sheet having positive charge density $\sigma_{\mathrm{o}}$. Choose the correct option from the following.

An electric dipole of mass $m$, charge $q$, and length $l$ is placed in a uniform electric field $\vec{E} = E_0\hat{i}$. When the dipole is rotated slightly from its equilibrium position and released, the time period of its oscillations will be :

Match List - I with List - II.

Choose the correct answer from the options given below :

Three infinitely long wires with linear charge density $\lambda$ are placed along the $x-a x i s, y-a x i s$ and $z-$ axis respectively. Which of the following denotes an equipotential surface?

A particle of mass ' $m$ ' and charge ' $q$ ' is fastened to one end ' $A$ ' of a massless string having equilibrium length $l$, whose other end is fixed at point ' $O$ '. The whole system is placed on a frictionless horizontal plane and is initially at rest. If uniform electric field is switched on along the direction as shown in figure, then the speed of the particle when it crosses the $x$-axis is

A small uncharged conducting sphere is placed in contact with an identical sphere but having $4 \times 10^{-8} \mathrm{C}$ charge and then removed to a distance such that the force of repulsion between them is $9 \times 10^{-3} \mathrm{~N}$. The distance between them is (Take $\frac{1}{4 \pi \epsilon_{\mathrm{o}}}$ as $9 \times 10^9$ in SI units)

In the first configuration (1) as shown in the figure, four identical charges $\left(q_0\right)$ are kept at the corners A, B, C and D of square of side length ' $a$ '. In the second configuration (2), the same charges are shifted to mid points $G, E, H$ and $F$, of the square. If $K=\frac{1}{4 \pi \epsilon_0}$, the difference between the potential energies of configuration (2) and (1) is given by :

Consider a parallel plate capacitor of area A (of each plate) and separation ' $d$ ' between the plates. If $E$ is the electric field and $\varepsilon_0$ is the permittivity of free space between the plates, then potential energy stored in the capacitor is

Two point charges $-4 \mu \mathrm{c}$ and $4 \mu \mathrm{c}$, constituting an electric dipole, are placed at $(-9,0,0) \mathrm{cm}$ and $(9,0,0) \mathrm{cm}$ in a uniform electric field of strength $10^4 \mathrm{NC}^{-1}$. The work done on the dipole in rotating it from the equilibrium through $180^{\circ}$ is :

Two charges $7 \mu \mathrm{c}$ and $-4 \mu \mathrm{c}$ are placed at $(-7 \mathrm{~cm}, 0,0)$ and $(7 \mathrm{~cm}, 0,0)$ respectively. Given, $\epsilon_0=8.85 \times 10^{-12} \mathrm{C}^2 \mathrm{~N}^{-1} \mathrm{~m}^{-2}$, the electrostatic potential energy of the charge configuration is :

A point particle of charge $Q$ is located at $P$ along the axis of an electric dipole 1 at a distance $r$ as shown in the figure. The point P is also on the equatorial plane of a second electric dipole 2 at a distance r. The dipoles are made of opposite charge q separated by a distance $2 a$. For the charge particle at P not to experience any net force, which of the following correctly describes the situation?

The electric flux is $\phi=\alpha \sigma+\beta \lambda$ where $\lambda$ and $\sigma$ are linear and surface charge density, respectively. $\left(\frac{\alpha}{\beta}\right)$ represents

For a short dipole placed at origin O , the dipole moment P is along $x$-axis, as shown in the figure. If the electric potential and electric field at $A$ are $V_0$ and $E_0$, respectively, then the correct combination of the electric potential and electric field, respectively, at point B on the $y$-axis is given by

A line charge of length $\frac{\mathrm{a}}{2}$ is kept at the center of an edge $B C$ of a cube ABCDEFGH having edge length ' $a$ ' as shown in the figure. If the density of line charge is $\lambda \mathrm{C}$ per unit length, then the total electric flux through all the faces of the cube will be ___________ . (Take, $\epsilon_0$ as the free space permittivity)

Five charges $+q,+5 q,-2 q,+3 q$ and $-4 q$ are situated as shown in the figure. The electric flux due to this configuration through the surface $S$ is :

Two charged conducting spheres of radii $a$ and $b$ are connected to each other by a conducting wire. The ratio of charges of the two spheres respectively is:

Two identical conducting spheres P and S with charge Q on each, repel each other with a force $16 \mathrm{~N}$. A third identical uncharged conducting sphere $\mathrm{R}$ is successively brought in contact with the two spheres. The new force of repulsion between $\mathrm{P}$ and $\mathrm{S}$ is :

$\sigma$ is the uniform surface charge density of a thin spherical shell of radius R. The electric field at any point on the surface of the spherical shell is :

The vehicles carrying inflammable fluids usually have metallic chains touching the ground:

In hydrogen like system the ratio of coulombian force and gravitational force between an electron and a proton is in the order of :

A charge $q$ is placed at the center of one of the surface of a cube. The flux linked with the cube is:

An infinitely long positively charged straight thread has a linear charge density $\lambda \mathrm{~Cm}^{-1}$. An electron revolves along a circular path having axis along the length of the wire. The graph that correctly represents the variation of the kinetic energy of electron as a function of radius of circular path from the wire is :

Force between two point charges $q_1$ and $q_2$ placed in vacuum at '$r$' cm apart is $F$. Force between them when placed in a medium having dielectric constant $K=5$ at '$r / 5$' $\mathrm{cm}$ apart will be:

Two charges $q$ and $3 q$ are separated by a distance '$r$' in air. At a distance $x$ from charge $q$, the resultant electric field is zero. The value of $x$ is :

A particle of charge '$-q$' and mass '$m$' moves in a circle of radius '$r$' around an infinitely long line charge of linear charge density '$+\lambda$'. Then time period will be given as :

(Consider $k$ as Coulomb's constant)

The electrostatic potential due to an electric dipole at a distance '$r$' varies as :

An electric field is given by $(6 \hat{i}+5 \hat{j}+3 \hat{k}) \mathrm{N} / \mathrm{C}$. The electric flux through a surface area $30 \hat{i} \mathrm{~m}^2$ lying in YZ-plane (in SI unit) is :

Two charges of $5 Q$ and $-2 Q$ are situated at the points $(3 a, 0)$ and $(-5 a, 0)$ respectively. The electric flux through a sphere of radius '$4 a$' having center at origin is :

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

Assertion (A) : Work done by electric field on moving a positive charge on an equipotential surface is always zero.

Reason (R) : Electric lines of forces are always perpendicular to equipotential surfaces.

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

An electric charge $10^{-6} \mu \mathrm{C}$ is placed at origin $(0,0)$ $\mathrm{m}$ of $\mathrm{X}-\mathrm{Y}$ co-ordinate system. Two points $\mathrm{P}$ and $\mathrm{Q}$ are situated at $(\sqrt{3}, \sqrt{3}) \mathrm{m}$ and $(\sqrt{6}, 0) \mathrm{m}$ respectively. The potential difference between the points $\mathrm{P}$ and $\mathrm{Q}$ will be :

A $10 ~\mu \mathrm{C}$ charge is divided into two parts and placed at $1 \mathrm{~cm}$ distance so that the repulsive force between them is maximum. The charges of the two parts are:

Two charges each of magnitude $0.01 ~\mathrm{C}$ and separated by a distance of $0.4 \mathrm{~mm}$ constitute an electric dipole. If the dipole is placed in an uniform electric field '$\vec{E}$' of 10 dyne/C making $30^{\circ}$ angle with $\vec{E}$, the magnitude of torque acting on dipole is:

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

Assertion A : If an electric dipole of dipole moment $30 \times 10^{-5} ~\mathrm{C} ~\mathrm{m}$ is enclosed by a closed surface, the net flux coming out of the surface will be zero.

Reason R : Electric dipole consists of two equal and opposite charges.

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

In a metallic conductor, under the effect of applied electric field, the free electrons of the conductor

Electric potential at a point '$\mathrm{P}$' due to a point charge of $5 \times 10^{-9} \mathrm{C}$ is $50 \mathrm{~V}$. The distance of '$\mathrm{P}$' from the point charge is:

(Assume, $\frac{1}{4 \pi \varepsilon_{0}}=9 \times 10^{+9} ~\mathrm{Nm}^{2} \mathrm{C}^{-2}$ )

Graphical variation of electric field due to a uniformly charged insulating solid sphere of radius $\mathrm{R}$, with distance $r$ from the centre O is represented by:

A dipole comprises of two charged particles of identical magnitude $q$ and opposite in nature. The mass 'm' of the positive charged particle is half of the mass of the negative charged particle. The two charges are separated by a distance '$l$'. If the dipole is placed in a uniform electric field '$\bar{E}$'; in such a way that dipole axis makes a very small angle with the electric field, '$\bar{E}$'. The angular frequency of the oscillations of the dipole when released is given by:

For a uniformly charged thin spherical shell, the electric potential (V) radially away from the centre (O) of shell can be graphically represented as -

Let $\sigma$ be the uniform surface charge density of two infinite thin plane sheets shown in figure. Then the electric fields in three different region $E_{I}, E_{I I}$ and $E_{I I I}$ are:

Which of the following correctly represents the variation of electric potential $(\mathrm{V})$ of a charged spherical conductor of radius $(\mathrm{R})$ with radial distance $(\mathrm{r})$ from the center?

Electric field in a certain region is given by $\overrightarrow{\mathrm{E}}=\left(\frac{\mathrm{A}}{x^{2}} \hat{i}+\frac{\mathrm{B}}{y^{3}} \hat{j}\right) \text {. The } \mathrm{SI} \text { unit of } \mathrm{A} \text { and } \mathrm{B}$ are :

Two isolated metallic solid spheres of radii $\mathrm{R}$ and $2 \mathrm{R}$ are charged such that both have same charge density $\sigma$. The spheres are then connected by a thin conducting wire. If the new charge density of the bigger sphere is $\sigma^{\prime}$. The ratio $\frac{\sigma^{\prime}}{\sigma}$ is :

A point charge $2\times10^{-2}~\mathrm{C}$ is moved from P to S in a uniform electric field of $30~\mathrm{NC^{-1}}$ directed along positive x-axis. If coordinates of P and S are (1, 2, 0) m and (0, 0, 0) m respectively, the work done by electric field will be

In a cuboid of dimension $2 \mathrm{~L} \times 2 \mathrm{~L} \times \mathrm{L}$, a charge $q$ is placed at the center of the surface '$\mathrm{S}$' having area of $4 \mathrm{~L}^{2}$. The flux through the opposite surface to '$\mathrm{S}$' is given by

A point charge of 10 $\mu$C is placed at the origin. At what location on the X-axis should a point charge of 40 $\mu$C be placed so that the net electric field is zero at $x=2$cm on the X-axis?

The electric potential at the centre of two concentric half rings of radii R$_1$ and R$_2$, having same linear charge density $\lambda$ is :

If two charges q$_1$ and q$_2$ are separated with distance 'd' and placed in a medium of dielectric constant K. What will be the equivalent distance between charges in air for the same electrostatic force?

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

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

$\rho(r)= \begin{cases}\rho_{0}\left(\frac{3}{4}-\frac{r}{R}\right) & \text { for } r \leq R \\ \text { zero } & \text { for } r>R\end{cases}$

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.

A uniform electric field $\mathrm{E}=(8 \mathrm{~m} / \mathrm{e}) \,\mathrm{V} / \mathrm{m}$ is created between two parallel plates of length $1 \mathrm{~m}$ as shown in figure, (where $\mathrm{m}=$ mass of electron and e = charge of electron). An electron enters the field symmetrically between the plates with a speed of $2 \mathrm{~m} / \mathrm{s}$. The angle of the deviation $(\theta)$ of the path of the electron as it comes out of the field will be _________.

A charge of $4 \,\mu \mathrm{C}$ 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 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 $\mathrm{q}_{0}$ executes $\mathrm{SHM}$. The time period of oscillation of charge $\mathrm{q}_{0}$ will be :

Two uniformly charged spherical conductors $A$ and $B$ of radii $5 \mathrm{~mm}$ and $10 \mathrm{~mm}$ are separated by a distance of $2 \mathrm{~cm}$. 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 :

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 :

If the electric potential at any point (x, y, z) m in space is given by V = 3x² volt. The electric field at the point (1, 0, 3) m will be :

A positive charge particle of 100 mg is thrown in opposite direction to a uniform electric field of strength 1 $\times$ 10⁵ 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 A and B of magnitude +8 $\times$ 10^$-$6 C and $-$8 $\times$ 10^$-$6 C respectively are placed at a distance d apart. The electric field at the middle point O between the charges is 6.4 $\times$ 10⁴ NC^$-$1. The distance 'd' between the point charges A and B is :

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 :

If a charge q is placed at the centre of a closed hemispherical non-conducting surface, the total flux passing through the flat surface would be :

Three identical charged balls each of charge 2 C are suspended from a common point P by silk threads of 2 m each (as shown in figure). They form an equilateral triangle of side 1m.

The ratio of net force on a charged ball to the force between any two charged balls will be :

Sixty four conducting drops each of radius 0.02 m and each carrying a charge of 5 $\mu$C are combined to form a bigger drop. The ratio of surface density of bigger drop to the smaller drop will be :

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.

In the figure, a very large plane sheet of positive charge is shown. P₁ and P₂ are two points at distance l and 2l from the charge distribution. If $\sigma$ is the surface charge density, then the magnitude of electric fields E₁ and E₂ at P₁ and P₂ respectively are :

Two identical charged particles each having a mass 10 g and charge 2.0 $\times$ 10^$-$7C are placed on a horizontal table with a separation of L between them such that they stay in limited equilibrium. If the coefficient of friction between each particle and the table is 0.25, find the value of L. [Use g = 10 ms^$-$2]

A long cylindrical volume contains a uniformly distributed charge of density $\rho$. The radius of cylindrical volume is R. A charge particle (q) revolves around the cylinder in a circular path. The kinetic energy of the particle is :

A vertical electric field of magnitude 4.9 $\times$ 10⁵ 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 m/s²)

Statement- $1:$ When a charge $q$ is take from the centre of the surface of the sphere its potential energy changes by ${{q\rho } \over {3{\varepsilon _0}}}$
Statement- $2:$ The electric field at a distance $r\left( {r < R} \right)$ from the center of the sphere is ${{\rho r} \over {3{\varepsilon _0}}}.$

Statement-1 : For a charged particle moving from point $P$ to point $Q$, the net work done by an electrostatic field on the particle is independent of the path connecting point $P$ to point $Q.$
Statement-2 : The net work done by a conservative force on an object moving along a closed loop is zero.