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

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

A uniform electric field of \( 10 \, \text{N/C} \) is created between two parallel charged plates (as shown in figure). An electron enters the field symmetrically between the plates with a kinetic energy \( 0.5 \, \text{eV} \). the length of each plate is \( 10 \, \text{cm} \). The angle (\( \theta \)) of deviation of the path of electron as it comes out of the field is ________ (in degree).

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:

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

A thin infinite sheet charge and an infinite line charge of respective charge densities \( +\sigma \) and \( +\lambda \) are placed parallel at \( 5 \, \text{m} \) distance from each other. Points ‘P’ and ‘Q’ are at \( \dfrac{3}{\pi} \, \text{m} \) and \( \dfrac{4}{\pi} \, \text{m} \) perpendicular distances from line charge towards sheet charge, respectively. ‘\( E_P \)’ and ‘\( E_Q \)’ are the magnitudes of resultant electric field intensities at point ‘P’ and ‘Q’, respectively. If \( \dfrac{E_P}{E_Q} = \dfrac{4}{a} \) for \( 2|\sigma| \neq |\lambda| \), then the value of \( a \) is ______

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason (R).
Assertion A: Two metallic spheres are charged to the same potential. One of them is hollow and another is solid, and both have the same radii. Solid sphere will have lower charge than the hollow one.
Reason R: Capacitance of metallic spheres depend on the radii of spheres.
In the light of the above statements, choose the correct answer from the options given below:

Two charges each of magnitude \( 0.01 \, \text{C} \) and separated by a distance of \( 0.4 \, \text{mm} \) constitute an electric dipole. If the dipole is placed in an uniform electric field ‘\( \vec{E} \)’ of \( 10 \, \text{dyne}/\text{C} \) 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 \(30 \times 10^{-5} \, \text{C m}\) 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.

Three concentric spherical metallic shells X, Y and Z of radius \( a \), \( b \) and \( c \) respectively \([ a < b < c ]\) have surface charge densities \( \sigma \), \( -\sigma \) and \( \sigma \), respectively. The shells X and Z are at same potential. If the radii of X and Y are \( 2 \, \text{cm} \) and \( 3 \, \text{cm} \), respectively. The radius of shell Z is ________ \( \text{cm} \).

Three point charges \( q \), \( -2q \) and \( 2q \) are placed on x-axis at a distance \( x=0 \), \( x=\dfrac{3}{4}R \) and \( x=R \) respectively from origin as shown. If \( q=2\times 10^{-6} \, \text{C} \) and \( R=2 \, \text{cm} \), the magnitude of net force experienced by the charge \( -2q \) is __________ \( \text{N} \).

As shown in figure, a cuboid lies in a region with electric field \( {E} = 2x^{2}\hat{i} - 4y\hat{j} + 6\hat{k} \, \frac{\text{N}}{\text{C}} \). The magnitude of charge within the cuboid is \( n \varepsilon_{0} \, \text{C} \). The value of \( n \) is _____. (If dimension of cuboid is \( 1 \times 2 \times 3 \, \text{m}^{3} \))

In a cuboid of dimension \( 2L \times 2L \times L \), a charge \( q \) is placed at the center of the surface ‘S’ having area of \( 4L^{2} \). The flux through the opposite surface to ‘S’ is given by:

Considering a group of positive charges, which of the following statements is correct?

If two charges \( q_1 \) & \( q_2 \) 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 equal positive point charges are separated by a distance \( 2a \). The distance of a point from the centre of the line joining two charges on the equatorial line (perpendicular bisector) at which force experienced by a test charge \( q_0 \) becomes maximum is \( \frac{a}{\sqrt{x}} \). The value of \( x \) is ________.

Two isolated metallic solid spheres of radii \( R \) and \( 2R \) 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 \( \sigma' \). The ratio \( \frac{\sigma'}{\sigma} \) is:

Electric potential at a point ‘P’ due to a point charge of \( 5 \times 10^{-9} \, \text{C} \) is \( 50 \, \text{V} \). The distance of ‘P’ from the point charge is:
(Assume, \( \dfrac{1}{4\pi \varepsilon_{0}} = 9 \times 10^{+9} \, \text{N m}^{2} \text{C}^{-2} \) )

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

A capacitor of capacitance \( 900 \, \mu \text{F} \) charged by a \( 100 \, \text{V} \) battery. The capacitor is disconnected from the battery and connected to another uncharged identical capacitor such that one plate of uncharged capacitor connected to positive plate and another plate of uncharged connected to negative plate of the charged capacitor. The loss of energy in this process is measured as \( x \times 10^{-2} \, \text{J} \). The value of \( x \) is ________.

As shown in the figure, a configuration of two equal point charges \( (q_0 = +2 \, \mu\text{C}) \) is placed on an inclined plane. Mass of each point charge is \( 20 \, \text{g} \). Assume that there is no friction between charge and plane. For the system of two point charges to be in equilibrium (at rest) the height \( h = x \times 10^{-3} \, \text{m} \). The value of \( x \) is ______.
(Take \( \dfrac{1}{4\pi \varepsilon_0} = 9 \times 10^9 \, \text{N m}^2 \text{C}^{-2} \), \( g = 10 \, \text{m s}^{-2} \))

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:

For a charged spherical ball, electrostatic potential inside the ball varies with \( r \) as \( V = 2ar^2 + b \).
Here, \( a \) and \( b \) are constant; \( r \) is the distance from the centre. The volume charge density inside the ball is \( -\lambda a \varepsilon \). The value of \( \lambda \) is _____________.
\( \varepsilon \) = permittivity of the medium.

Expression for an electric field is given by \( \vec{E} = 4000 x^{2} \hat{i} \, \dfrac{\text{V}}{\text{m}} \). The electric flux through the cube of side \( 20 \, \text{cm} \) when placed in electric field (as shown in the figure) is _____ \( \text{V cm} \).

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 \( {E}_{I} \), \({E}_{II} \) and \( {E}_{III} \) are:

A cubical volume is bounded by the surfaces \( x = 0 \), \( x = a \), \( y = 0 \), \( y = a \), \( z = 0 \), \( z = a \). The electric field in the region is given by \( \vec{E} = E_{0}x\,\hat{i} \). Where \( E_{0} = 4 \times 10^{4} \,\text{N C}^{-1}\text{m}^{-1} \). If \( a = 2 \,\text{cm} \), the charge contained in the cubical volume is \( Q \times 10^{-14} \,\text{C} \). The value of \( Q \) is _______.

An electron revolves around an infinite cylindrical wire having uniform linear charge density \( 2 \times 10^{-8} \, \text{C m}^{-1} \) in circular path under the influence of attractive electrostatic field as shown in the figure. The velocity of electron with which it is revolving is ______ \( \times 10^{6} \, \text{m s}^{-1} \).
Given mass of electron \( = 9 \times 10^{-31} \, \text{kg} \).

An electric dipole of dipole moment is \( 6.0 \times 10^{-6} \, \text{C m} \) placed in a uniform electric field of \( 1.5 \times 10^{3} \, \text{N C}^{-1} \) in such a way that dipole moment is along electric field. The work done in rotating dipole by \( 180^\circ \) in this field will be ______ \( \text{mJ} \).

A point charge \( q_1 = 4q_0 \) is placed at origin. Another point charge \( q_2 = -q_0 \) is placed at \( x = 12 \, \text{cm} \). Charge of proton is \( q_0 \). The proton is placed on \( x \) axis so that the electrostatic force on the proton is zero. In this situation, the position of the proton from the origin is ______ \( \text{cm} \).

A stream of a positively charged particles having \( \frac{q}{m} = 2 \times 10^{11} \, \frac{\mathrm{C}}{\mathrm{kg}} \) and velocity \( \vec{v}_0 = 3 \times 10^{7} \, \hat{i} \, \frac{\mathrm{m}}{\mathrm{s}} \) is deflected by an electric field \( 1.8 \, \hat{j} \, \mathrm{kV}/\mathrm{m} \). The electric field exists in a region of \( 10 \, \mathrm{cm} \) along \( x \) direction. Due to the electric field, the deflection of the charge particles in the \( y \) direction is ______.

point charge \( 2 \times 10^{-2} \, \text{C} \) is moved from P to S in a uniform electric field of \( 30 \, \text{N} \, \text{C}^{-1} \) directed along positive x-axis. If coordinates of P and S are \( (1, \, 2, \, 0) \, \text{m} \) and \( (0, \, 0, \, 0) \, \text{m} \) respectively, the work done by electric field will be:

\( 64 \) identical drops each charged upto potential of \( 10 \, \text{mV} \) are combined to form a bigger drop. The potential of the bigger drop will be ______ \( \text{mV} \).

The electric field due to a short electric dipole at a large distance \( (r) \) from center of dipole on the equatorial plane varies with distance as:

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

As shown in the figure, a point charge \( Q \) is placed at the centre of conducting spherical shell of inner radius \( a \) and outer radius \( b \). The electric field due to charge \( Q \) in three different regions \( I \), \( II \) and \( III \) is given by: \( ( I : r < a , \, II : a < r < b , \, III : r > b ) \).

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JEE Main 2023 1st Feb Morning Shift

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 \( {E}_{I} \), \({E}_{II} \) and \( {E}_{III} \) are: