Electrostatics — JEE Main PYQs

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 $1.5 \text{ g/cm}^3$) droplets of radius 1 mm horizontally. All oil droplets carry a charge 5 nC. The potentials $V_A$, $V_B$ 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 $g = 10 \text{ m/s}^2$)

Two point charges $8\ \mu\text{C}$ and $-2\ \mu\text{C}$ 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 ________.

A parallel plate air capacitor is connected to a battery. The plates are pulled apart at uniform speed $v$. If $x$ is the separation between the plates at any instant, then the time rate of change of electrostatic energy of the capacitor is proportional to $x^{\alpha}$, where $\alpha$ 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)?

From the circuit given below, the capacitance between terminals $A$ and $B$ shown in the circuit is __________ $\mu\text{F}$. (take $C_1 = C_2 = C_3 = 1\ \mu\text{F}$ and $C_4 = 2\ \mu\text{F}$.)

[Figure: A capacitor network between terminals A and B: C1, C2 and C3 in series along the line from A to B; a wire bypasses C1 (connecting A-side to the node between C1 and C2 via the top), and C4 is connected in a branch across C2 and C3 at the bottom.]

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

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.

($\epsilon_o = 8.85 \times 10^{-12}\ \text{C}^2/\text{Nm}^2$ 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.

A sphere of capacitance 100 pF is charged to a potential of 100 V. Another identical uncharged metal sphere is brought in contact with the charged sphere, then the change in the total energy stored on these spheres, when they touch is $\alpha \times 10^{-7}$ J. The value of $\alpha$ is __________. (combined capacitance of spheres is 200 pF)

Two point charges $q_1 = 3\,\mu\text{C}$ and $q_2 = -4\,\mu\text{C}$ 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. $\left(\text{Take } \dfrac{1}{4\pi\epsilon_0} = 9 \times 10^9 \text{ SI Units}\right)$

A parallel plate capacitor is having separation between plates $0.885\,\text{mm}$. It has a capacitance of $1\,\mu\text{F}$ when the space between the plates is filled with an insulating material of resistivity $1 \times 10^{13}\,\Omega\text{m}$ and resistance $17.7 \times 10^{14}\,\Omega$. Relative permittivity of the insulating material is $\alpha \times 10^7$. The value of $\alpha$ is ________.

(Take permittivity of free space $= 8.85 \times 10^{-12}\,\text{F/m}$)

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 $45°$ with the sheet at equilibrium the charge density of sheet will be.

(Given, $\epsilon_0=8.85\times10^{-12}~\frac{\mathrm{F}}{\mathrm{m}}$ and acceleration due to gravity, $g=10~\frac{\mathrm{m}}{\mathrm{s}^2}$)

[Figure: a vertical positively charged sheet with charge density $+\sigma$; a string of length $l=1$ m attached at point A on the sheet makes $45°$ with the sheet, with a bob of charge $+10~\mu\mathrm{C}$ at its end]

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 :

[Figure: two parallel plates separated by distance d, with charge densities $+\sigma$ (left) and $-2\sigma$ (right); point charge $+q$ at the midpoint, d/2 from each plate]

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

[Figure: two vertical parallel plates; between them a right triangle ACB with C at the base of the left region, vertical side CA = 3 cm, hypotenuse AB = 5 cm, and B on the base to the right of C; the base CB is parallel to the plates' separation direction]

Consider a circular loop that is uniformly charged and has a radius $a\sqrt{2}$. 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 $xy$ plane at the origin :

A parallel plate capacitor is filled equally(half) with two dielectrics of dielectric constants $\varepsilon_1$ and $\varepsilon_2$, as shown in figures. The distance between the plates is d and area of each plate is A. If capacitance in first configuration and second configuration are $C_1$ and $C_2$ respectively, then $\frac{C_1}{C_2}$ is :

First Configuration

[Figure: parallel plate capacitor with the two dielectric slabs stacked horizontally — $\varepsilon_2$ occupying the upper half and $\varepsilon_1$ the lower half of the gap (dielectrics in series)]

Second Configuration

[Figure: parallel plate capacitor with the two dielectric slabs side by side — $\varepsilon_2$ filling the left half and $\varepsilon_1$ the right half of the gap (dielectrics in parallel)]

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

Using a battery, a 100 pF capacitor is charged to 60 V and then the battery is removed. After that, a second uncharged capacitor is connected to the first capacitor in parallel. If the final voltage across the second capacitor is 20 V, its capacitance is: (in pF)

Three parallel plate capacitors $\mathrm{C_1}$, $\mathrm{C_2}$ and $\mathrm{C_3}$ each of capacitance $5 \mathrm{~\mu F}$ are connected as shown in figure. The effective capacitance between points A and B, when the space between the parallel plates of $\mathrm{C_1}$ capacitor is filled with a dielectric medium having dielectric constant of 4, is :

[Figure: between A and B, C1 and C2 in series form the top branch; C3 alone forms the bottom branch in parallel with it]

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

[Figure: a uniformly charged ring (charges marked ×) with centre O; A at top, B at right, C at bottom, D at left; AC and BD are perpendicular diameters]

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:

[Figure: charge $q_1$ at point A, charge $q_2$ at point B 30 cm to its right; charge $q_3$ at point C, 40 cm vertically above A; a quarter-circle arc of radius 40 cm centred at A from C down to point D on the line AB beyond B]

A dipole with two electric charges of 2 $\mu$C magnitude each, with separation distance 0.5 $\mu$m, 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 air craft 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 :

A parallel plate capacitor has charge $5 \times 10^{-6}$ C. A dielectric slab is inserted between the plates and almost fills the space between the plates. If the induced charge on one face of the slab is $4 \times 10^{-6}$ C then the dielectric constant of the slab is __________.

The electric field in a region is given by $\vec{E} = \left(2\hat{i} + 4\hat{j} + 6\hat{k}\right) \times 10^3$ N/C. The flux of the field through a rectangular surface parallel to $x$-$z$ plane is $6.0\ \mathrm{Nm^2C^{-1}}$. The area of the surface is __________ $\mathrm{cm^2}$.

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

An infinitely long wire has uniform linear charge density $\lambda = 2\ \mathrm{nC/m}$. 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 $x\ \mathrm{Nm^2C^{-1}}$, where $x$ is:

[Neglect any edge effects and use $\frac{1}{4\pi\varepsilon_0} = 9 \times 10^9$ SI units]

Electric charge is transferred to an irregular metallic disk as shown in 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:

[Figure: an irregular metallic disk shaped like a pointed oval/eye — sharp points at top ($\sigma_1$) and bottom ($\sigma_3$), broader flat sides at right ($\sigma_2$) and left ($\sigma_4$)]

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

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

[Figure: parallel plate capacitor with the gap split into two equal horizontal layers, each of thickness $\frac{d}{2}$ — the upper layer has dielectric constant $k = 5$ and the lower layer has $k = 3$]

Space between the plates of a parallel plate capacitor of plate area 4 cm$^2$ and separation of (d) 1.77 mm, is filled with uniform dielectric materials with dielectric constants (3 and 5)as shown in figure. Another capacitor of capacitance 7.5 pF is connected in parallel with it. The effective capacitance of this combination is __________ pF.

(Given $\epsilon_0 = 8.85 \times 10^{-12}$ F/m)

A parallel-plate capacitor of capacitance $40\ \mu\mathrm{F}$ is connected to a 100 V power supply. Now the intermediate space between the plates is filled with a dielectric material of dielectric constant $K = 2$. Due to the introduction of dielectric material, the extra charge and the change in the electrostatic energy in the capacitor, respectively, are

An electron is made to enter symmetrically between two parallel and equally but oppositely charged metal plates, each of 10 cm length. The electron emerges out of the electric field region with a horizontal component of velocity $10^6$ m/s. If the magnitude of the electric field between the plates is 9.1 V/cm, then the vertical component of velocity of electron is

(mass of electron $= 9.1 \times 10^{-31}$ kg and charge of electron $= 1.6 \times 10^{-19}$ C)

A line charge of length $\frac{'a'}{2}$ is kept at the center of an edge BC of a cube ABCDEFGH having edge length 'a' as shown in the figure. If the density of line charge is $\lambda$ 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)

[Figure: cube ABCDEFGH of edge a with a short line charge centred on the bottom edge BC]

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

[Figure: dipole P at origin O pointing along the positive $x$-axis; point A on the $x$-axis at distance r from O; point B on the $y$-axis at distance 2r from O]

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

[Figure: dipole 1 (charges $+q$ and $-q$ separated by $2a$) lies along a horizontal axis with point P on this axis at distance $r$ from it; dipole 2 (charges $+q$ and $-q$ separated by $2a$) lies horizontally below P, with P at distance $r$ on its equatorial (perpendicular bisector) line]

A positive ion A and a negative ion B has charges $6.67 \times 10^{-19}$ C and $9.6 \times 10^{-10}$ C, and masses $19.2 \times 10^{-27}$ kg and $9 \times 10^{-27}$ kg respectively. At an instant, the ions are separated by a certain distance. At that instant the ratio of the magnitudes of electrostat_____

[Stem truncated: the remainder of this question is lost due to PDF corruption; the surviving text ends mid-sentence]

Two point charges $-4~\mu c$ and $4~\mu c$, constituting an electric dipole, are placed at $(-9, 0, 0)$ cm and $(9, 0, 0)$ 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 $180°$ is :

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

A parallel plate capacitor was made with two rectangular plates, each of length $l = 3$ cm and breadth $b = 1$ cm. The distance between the plates is 3 $\mu\mathrm{m}$. Out of the following, which are the ways to increase the capacitance by a factor of 10?

A. $l = 30$ cm, $b = 1$ cm, $d = 1~\mu\mathrm{m}$

B. $l = 3$ cm, $b = 1$ cm, $d = 30~\mu\mathrm{m}$

C. $l = 6$ cm, $b = 5$ cm, $d = 3~\mu\mathrm{m}$

D. $l = 1$ cm, $b = 1$ cm, $d = 10~\mu\mathrm{m}$

E. $l = 5$ cm, $b = 2$ cm, $d = 1~\mu\mathrm{m}$

Choose the correct answer from the options given below:

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

A square loop of sides $a = 1$ m is held normally in front of a point charge $q = 1\mathrm{C}$. The flux of the electric field through the shaded region is $\frac{5}{p}\times\frac{1}{\varepsilon_0}~\frac{\mathrm{Nm^2}}{\mathrm{C}}$, where the value of p is _____.

[Figure: a square of side $a$ divided into four quadrants; the left half of the top-left quadrant (triangular region) and the bottom two quadrants are shaded; the point charge q lies on the axis through the centre of the square]

A small uncharged conducting sphere is kept in contact with an identical sphere having charge $4 \times 10^{-8}\,\mathrm{C}$ 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 $\frac{1}{4\pi\epsilon_0}$ as $9 \times 10^{9}$ in SI units)

[Figure: two squares of side 'a'. Configuration (1): identical charges $+q_0$ at the four corners A, B, C, D; G, E, H, F are the midpoints of sides AB, BC, DC and AD respectively, with centre O. Configuration (2): the same charges $+q_0$ placed at the midpoints G, E, H and F]

In the first configuration (1) as shown in the figure, four identical charges $(q_0)$ 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 :

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 a 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

[Figure: x-y axes with origin at O; a uniform electric field E points along the +x direction; the string OA of length $l$ makes an angle of 60° with the x-axis, with the particle at end A]

Two capacitors $C_1$ and $C_2$ are connected in parallel to a battery. The charge-time graph for both the capacitors is shown below. The energies stored in them are $U_1$ and $U_2$ respectively. Which of the given statements is true?

[Figure: graph of charge q vs time t showing two rising curves — curve $C_1$ rises faster initially and saturates at a lower charge; curve $C_2$ rises slower but saturates at a higher charge]

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

A parallel plate capacitor of capacitance $1\,\mu\mathrm{F}$ is charged to a potential difference of 20 V. The distance between plates is $1\,\mu\mathrm{m}$. The energy density between plates of capacitor is.

An electric dipole of dipole moment $6 \times 10^{-6}\,\mathrm{Cm}$ is placed in uniform electric field of magnitude $10^6\,\mathrm{V/m}$. Initially, the dipole moment is parallel to electric field. The work that needs to be done on the dipole to make its dipole moment opposite to the field, will be __________ J.

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 :

An infinite plane sheet of charge having uniform surface charge density $+\sigma_s~\mathrm{C/m^2}$ is placed on $x$-$y$ plane. Another infinitely long line charge having uniform linear charge density $+\lambda_e~\mathrm{C/m}$ is placed at $z = 4$ m plane and parallel to $y$-axis. If the magnitude values $|\sigma_s| = 2|\lambda_e|$ then at point (0, 0, 2), the ratio of magnitudes of electric field values due to sheet charge to that of line charge is $\pi\sqrt{n} : 1$. The value of n is ________.

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

A parallel plate capacitor of capacitance $12.5~pF$ is charged by a battery connected between its plates to potential difference of 12.0 V. The battery is now disconnected and a dielectric slab ($\epsilon_r = 6$) is inserted between the plates. The change in its potential energy after inserting the dielectric slab is ______ $\times 10^{-12}$ J.

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

Three capacitors of capacitances $25\ \mu\mathrm{F}$, $30\ \mu\mathrm{F}$ and $45\ \mu\mathrm{F}$ are connected in parallel to a supply of 100 V. Energy stored in the above combination is E. When these capacitors are connected in series to the same supply, the stored energy is $\frac{9}{x}$ E. The value of $x$ is ________.

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

The electric field at point p due to an electric dipole is E. The electric field at point R on equitorial line will be $\frac{E}{x}$. The value of $x$ :

[Figure: a dipole with $-q$ and $+q$ on a horizontal axis about origin O; point p lies on the axial line at distance r from O; point Q lies on the equatorial (perpendicular) line at distance r from O and point R at distance 2r from O]

$\sigma$ 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 :

Three infinitely long charged thin sheets are placed as shown in figure. The magnitude of electric field at the point $P$ is $\frac{x\sigma}{\epsilon_o}$. The value of $x$ is _________ (all quantities are measured in SI units).

[Figure: three vertical sheets parallel to the YZ plane — sheet with charge density $-\sigma$ at $X=-a$, sheet with $-2\sigma$ at $X=a$, sheet with $\sigma$ at $X=3a$; point $P$ lies between $X=a$ and $X=3a$]

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

A capacitor of 10 μF capacitance whose plates are separated by 10 mm through air and each plate has area $4\ \mathrm{cm^2}$ is now filled equally with two dielectric media of $K_1 = 2$, $K_2 = 3$ respectively as shown in figure. If new force between the plates is 8 N. The supply voltage is ________ V.

[Figure: parallel-plate capacitor of plate separation d, left half filled with dielectric $K_1 = 2$ and right half with $K_2 = 3$, side by side]

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:

An electric field, $\vec{E} = \frac{2\hat{i} + 6\hat{j} + 8\hat{k}}{\sqrt{6}}$ passes through the surface of $4\ \mathrm{m^2}$ area having unit vector $\hat{n} = \left(\frac{2\hat{i} + \hat{j} + \hat{k}}{\sqrt{6}}\right)$. The electric flux for that surface is ______ V m.

A capacitor has air as dielectric medium and two conducting plates of area $12\ \mathrm{cm}^2$ and they are $0.6\ \mathrm{cm}$ apart. When a slab of dielectric having area $12\ \mathrm{cm}^2$ and $0.6\ \mathrm{cm}$ thickness is inserted between the plates, one of the conducting plates has to be moved by $0.2\ \mathrm{cm}$ to keep the capacitance same as in previous case. The dielectric constant of the slab is : (Given $\epsilon_0 = 8.834 \times 10^{-12}\ \mathrm{F/m}$)

If the net electric field at point P along Y axis is zero, then the ratio of $\left|\frac{q_2}{q_3}\right|$ is $\frac{8}{5\sqrt{x}}$, where $x =$ __________.

[Figure: point P lies 4 cm above the base line; charge $+q_2$ is on the base line 2 cm left of the foot of the perpendicular from P, and charge $-q_3$ is 3 cm right of it]

A capacitor is made of a flat plate of area A and a second plate having a stair-like structure as shown in figure. If the area of each stair is $\dfrac{A}{3}$ and the height is d, the capacitance of the arrangement is :

[Figure: a flat bottom plate of area A; above it a stair-like top plate with three steps each of area A/3, at heights d, 2d and 3d from the bottom plate]

At the centre of a half ring of radius $R = 10$ cm and linear charge density $4\ \mathrm{nC\ m^{-1}}$, the potential is $x\pi$ V. The value of $x$ is ________.

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

[Figure: an irregular closed surface S enclosing charges $+5q$, $q$ and $-2q$; charges $+3q$ and $-4q$ lie outside the surface]

An electric field $\vec{E} = (2x\hat{i})\,\mathrm{NC^{-1}}$ exists in space. A cube of side 2m is placed in the space as per figure given below. The electric flux through the cube is ________ $\mathrm{Nm^2/C}$.

[Figure: a cube of side 2 m placed with one face in the y–z plane at a distance 2 m from the origin along the x-axis]

Two identical capacitors have same capacitance C. One of them is charged to the potential V and other to the potential 2V. The negative ends of both are connected together. When the positive ends are also joined together, the decrease in energy of the combined system is :

Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle $\theta$ with each other. When suspended in water the angle remains the same. If density of the material of the sphere is 1.5 g/cc, the dielectric constant of water will be ________ (Take density of water = 1 g/cc)

$\mathrm{C_1}$ and $\mathrm{C_2}$ are two hollow concentric cubes enclosing charges 2Q and 3Q respectively as shown in figure. The ratio of electric flux passing through $\mathrm{C_1}$ and $\mathrm{C_2}$ is :

[Figure: a small cube containing charge 2Q placed inside a larger cube; charge 3Q lies inside the larger cube but outside the smaller one]

Suppose a uniformly charged wall provides a uniform electric field of $2 \times 10^4$ N/C normally. A charged particle of mass 2 g being suspended through a silk thread of length 20 cm and remain stayed at a distance of 10 cm from the wall. Then the charge on the particle will be $\frac{1}{\sqrt{x}}$ $\mu$C where $x =$ ________. [use $g = 10$ m/s$^2$]

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

A thin metallic wire having cross sectional area of $10^{-4}\ \mathrm{m^2}$ is used to make a ring of radius 30 cm. A positive charge of $2\pi$ C is uniformly distributed over the ring, while another positive charge of 30 pC is kept at the centre of the ring. The tension in the ring is ________ N; provided that the ring does not get deformed (neglect the influence of gravity).

$\left(\text{given, } \frac{1}{4\pi\epsilon_0} = 9 \times 10^9\ \text{SI units}\right)$

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

An electron is moving under the influence of the electric field of a uniformly charged infinite plane sheet S having surface charge density $+\sigma$. The electron at $t=0$ is at a distance of 1 m from S and has a speed of 1 m/s. The maximum value of $\sigma$ if the electron strikes S at $t=1$ s is $\alpha\left[\frac{m \epsilon_0}{e}\right] \frac{\mathrm{C}}{\mathrm{m}^2}$, the value of $\alpha$ is ______.

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

A capacitor of capacitance C and potential V has energy E. It is connected to another capacitor of capacitance 2 C and potential 2 V. Then the loss of energy is $\frac{x}{3}\,\mathrm{E}$, where $x$ 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)

Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle of $37°$ with each other. When suspended in a liquid of density $0.7\ g/cm^3$, the angle remains same. If density of material of the sphere is $1.4\ g/cm^3$, the dielectric constant of the liquid is _______ $\left(\tan 37° = \frac{3}{4}\right)$

Two charges $q$ and $3q$ 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 :