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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:

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

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:

Match List - I with List - II.

Choose the correct answer from the options given below:

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

An electric dipole of dipole moment \( 6 \times 10^{-6} \, \text{cm} \) is placed in uniform electric field of magnitude \( 10^{6} \, \text{V/m} \). 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 __________ \( \text{J} \).

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?

Two large plane parallel conducting plates are kept \( 10 \, \text{cm} \) 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:

Two small spherical balls of mass \( 10\,\text{g} \) each with charges \( -2\,\mu\text{C} \) and \( 2\,\mu\text{C} \), are attached to two ends of very light rigid rod of length \( 20\,\text{cm} \). The arrangement is now placed near an infinite non-conducting charge sheet with uniform charge density of \( 100\,\mu\text{C}/\text{m}^{2} \) 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 \( \varepsilon_{0}:8.85\times10^{-12}\,\text{C}^{2}/\text{Nm}^{2} \) )

Two charges \( q_1 \) and \( q_2 \) are separated by a distance of \( 30 \, \text{cm} \). A third charge \( q_3 \) initially at ‘C’ as shown in the figure, is moved along the circular path of radius \( 40 \, \text{cm} \) C to D. If the difference in potential energy due to movement of \( q_3 \) from C to D is given by \( \dfrac{q_3 K}{4 \pi \varepsilon_0} \), the value of \( K \) is:

A dipole with two electric charges of \( 2 \, \mu \text{C} \) magnitude each, with separation distance \( 0.5 \, \mu \text{m} \), 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 \( 5 \, \text{V} \) is applied. Separation between the plates is \( 0.5 \, \text{mm} \). If the dipole is rotated by \( 30^\circ \) from the axis, it tends to realign in the direction due to a torque. The value of torque is:

Two charges \( 7 \,\mu \text{C} \) and \( -4 \,\mu \text{C} \) are placed at \( (-7 \, \text{cm}, 0, 0) \) and \( (7 \, \text{cm}, 0, 0) \) respectively. Given, \( \varepsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2 \, \text{N}^{-1} \, \text{m}^{-2} \), the electrostatic potential energy of the charge configuration is:

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 \( A, B, C \) and \( D \) points is:

A small uncharged conducting sphere is placed in contact with an identical sphere but having \( 4 \times 10^{-8} \, \text{C} \) charge and then removed to a distance such that the force of repulsion between them is \( 9 \times 10^{-3} \, \text{N} \). The distance between them is (Take \( \frac{1}{4\pi \varepsilon_{0}} \) as \( 9 \times 10^{9} \) in SI units)

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 \( 10^4 \,\mathrm{N\,C^{-1}} \). The work done on the dipole in rotating it from the equilibrium through \( 180^\circ \) is:

A small bob of mass \( 100 \, \mathrm{mg} \) and charge \( +10 \, \mu\mathrm{C} \) is connected to an insulating string of length \( 1 \, \mathrm{m} \). 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, \( \varepsilon_{0} = 8.85 \times 10^{-12} \, \mathrm{} \) and acceleration due to gravity, \( g = 10 \, \mathrm{m/s^{2}} \) )

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:

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 electron is made to enter symmetrically between two parallel and equally but oppositely charged metal plates, each of \( 10 \,\text{cm} \) length. The electron emerges out of the electric field region with a horizontal component of velocity \( 10^{6} \,\text{m/s} \). If the magnitude of the electric field between the plates is \( 9.1 \,\text{V/cm} \), then the vertical component of velocity of electron is:
(mass of electron \( = 9.1 + 10^{-31} \,\text{kg} \) and charge of electron \( = 1.6 \times 10^{-19} \,\text{C} \))

A positive ion \( A \) and a negative ion \( B \) has charges \( 6.67 \times 10^{-19} \, \text{C} \) and \( 9.6 \times 10^{-10} \, \text{C} \), and masses \( 19.2 \times 10^{-27} \, \text{kg} \) and \( 9 \times 10^{-27} \, \text{kg} \) respectively. At an instant, the ions are separated by a certain distance \( r \). At that instant the ratio of the magnitudes of electrostatic force to gravitational force is \( P \times 10^{-13} \), where the value of \( P \) is __________.
(Take \( \dfrac{1}{4\pi \varepsilon_0} = 9 \times 10^{9} \, \text{Nm}^2 \text{C}^{-1} \) and universal gravitational constant as \( 6.67 \times 10^{-11} \, \text{Nm}^2\text{kg}^{-2} \))

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:

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. \( \sigma_1 = \sigma_2 = \sigma_3 = \sigma_4 \)

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 person 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 line charge of length \( \dfrac{'a'}{2} \) 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 lien charge is \( \lambda \) \( \text{C} \) per unit length, then the total electric flux through all the faces of the cube will be _______. (Take, \( \varepsilon_{0} \) as the free space permittivity)

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:

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

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

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 \( 2a \). For the charge particle at \( P \) not to experience any net force, which of the following correctly describes the situation?

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:

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 \( ABC \)?

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 \(\displaystyle K = \frac{1}{4\pi \epsilon_0} \), the difference between the potential energies of configuration \( (2) \) and \( (1) \) is given by:

A point charge causes an electric flux of \( -2 \times 10^{4} \, \text{N m}^{2} \text{C}^{-1} \) to pass through a spherical Gaussian surface of \( 8.0 \, \text{cm} \) radius, centred on the charge. The value of the point charge is:
(Given \( \varepsilon_{0} = 8.85 \times 10^{-12} \, \text{C}^{2} \text{N}^{-1} \text{m}^{-2} \))

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:

An infinitely long wire has uniform linear charge density \( \lambda = 2 \, \text{nC}/\text{m} \). The net flux through a Gaussian cube of side length \( \sqrt{3} \, \text{cm} \), if the wire passes through any two corners of the cube, that are maximally displaced from each other, would be \( x \, \text{N} \, \text{m}^{2} \, \text{C}^{-1} \), where \( x \) is: [Neglect any edge effects and use \( \dfrac{1}{4\pi \varepsilon_{0}} = 9 \times 10^{9} \) SI units]

An electric dipole is placed at a distance of \( 2 \, \text{cm} \) from an infinite plane sheet having positive charge density \( \sigma_{0} \). Choose the correct option from the following.

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:

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JEE MAIN 2025 4th April Morning Shift

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 \( A, B, C \) and \( D \) points is: