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The electric field in a region is given by \( \vec{E} = \dfrac{2}{5} E_{0} \hat{i} + \dfrac{3}{5} E_{0} \hat{j} \) with \( E_{0} = 4.0 \times 10^{3} \, \dfrac{N}{C} \). The flux of this field through a rectangular surface area \( 0.4 \, \mathrm{m^{2}} \) parallel to the \( Y - Z \) plane is _______ \( \mathrm{N\,m^{2}\,C^{-1}} \).

What will be the magnitude of electric field at point \( O \) as shown in figure? Each side of the figure is \( l \) and perpendicular to each other?

A point charge of \( +12 \, \mu\text{C} \) is at a distance \( 6 \, \text{cm} \) vertically above the centre of a square of side \( 12 \, \text{cm} \) as shown in figure. The magnitude of the electric flux through the square will be ________ \( \times 10^{3} \, \text{N m}^2/\text{C} \).

A simple pendulum of '\(m\)',length '\(l\)' and charge '\(+q\)' suspended in the electric field produced by two conducting parallel plates as shown. The value of deflection of pendulum in equilibrium position will be: 

Two particles A and B having charges \( 20\,\mu\text{C} \) and \( -5\,\mu\text{C} \) respectively are held fixed with a separation of \( 5\,\text{cm} \). At what position a third charged particle should be placed so that it does not experience a net electric force?

Two identical tennis balls each having mass '\( m \)' and charge '\( q \)' are suspended from a fixed point by threads \(\sigma\) length '\( l \)'. What is the equilibrium separation when each thread makes a small angle '\( \theta \)' with the vertical? 

Two ideal electric dipoles \( A \) and \( B \), having their dipole moment \( p_1 \) and \( p_2 \) respectively are placed on a plane with their centres at \( O \) as shown in the figure. At point \( C \) on the axis of dipole \( A \), the resultant electric field is making an angle of \( 37^\circ \) with the axis.
The ratio of the dipole moment of \( A \) and \( B \), \( \dfrac{p_1}{p_2} \) is : \( \left(\text{Take } \sin 37^\circ C= \dfrac{3}{5}\right) \). 

A cube is placed inside an electric field, \( \vec{E}=150y^{2}\,\hat{j} \). The side of the cube is \( 0.5 \, \text{m} \) and is placed in the field as shown in the given figure. The charge inside the cube is:

A body having specific charge \(8\,\mu\text{C}/g\) is resting on a frictionless plane at a distance \(10\,\text{cm}\) from the wall (as sown in the figure). It starts moving towards the wall when a uniform electric field of \(100\,\text{V/m}\) is applied horizontally towards the wall. If the collision of the body with the wall is perfectly elastic, then the time period of the motion will be ____________ \(\text{s}\).

A solid metal sphere of radius $R$ having charge $q$ is enclosed inside the concentric spherical shell of inner radius $a$ and outer radius $b$ as shown in the figure. The approximate variation electric field $\vec{E}$ as a function of distance $r$ from centre $O$ is given by

An infinite number of point charges, each carrying \( 1 \, \mu \text{C} \) charge, are placed along the y-axis at \( y = 1 \, \text{m}, 2 \, \text{m}, 4 \, \text{m}, 8 \, \text{m} \ldots \). The total force on a \( 1 \, \text{C} \) point charge, placed at the origin is \( x \times 10^{3} \, \text{N} \). The value of \( x \), to the nearest integer is:
[Take \( \dfrac{1}{4 \pi \epsilon_{0}} = 9 \times 10^{9} \, \text{N m}^{2} / \text{C}^{2} \)]

An electric dipole is placed on \( x \)-axis in proximity to a line charge of linear charge density \( 3.0 \times 10^{-6} \, \text{C/m} \). Line charge is placed on \( z \)-axis and positive and negative charge of dipole is at a distance of \( 10 \, \text{mm} \) and \( 12 \, \text{mm} \) from the origin respectively. If total force of \( 4 \, \text{N} \) is exerted on the dipole, find out the amount of positive or negative charge of the dipole.

A certain charge \( Q \) is divided into two part \( q \) and \( (Q - q) \). How should the charge \( Q \) and \( q \) be divided so that \( q \) and \( (Q - q) \) placed at a certain distance apart experience maximum electrostatic repulsion?

Find out the surface charge density at the intersection of point \( x = 3 \, \text{m} \) plane and x-axis, in the region of uniform line charge of \( 8 \, \text{nC/m} \) lying along the z-axis in free space.

Choose the incorrect statement:
(a) The electric lines of force entering into a Gaussian surface provide negative flux.
(b) A charge '\( q \)' is placed at the centre of a cube. The flux through all the faces will be the same.
(c) In a uniform electric field net flux through a closed Gaussian surface containing no net charge, is zero.
(d) When electric field is parallel to a Gaussian surface, it provides a finite non-zero flux.
Choose the most appropriate answer from the options given below:

The two thin coaxial rings, each of radius \( a \) and having charges \( +Q \) and \( -Q \) respectively are separated by a distance of \( s \). The potential difference between the centres of the two rings is:

The total charge enclosed in an incremental volume of \( 2 \times 10^{-9} \, \mathrm{m^{3}} \) located at the origin is ______ \( \mathrm{nC} \), if electric flux density of its field is found as:
\( \mathbf{D} = e^{-x} \sin y \, \hat{\imath} - e^{-x} \cos y \, \hat{\jmath} + 2 z \, \hat{k} \; \mathrm{C/m^{2}} \).

A uniformly charged disc of radius \( R \) having surface charge density \( \sigma \) is placed in the \( xy \) plane with its centre at the origin. Find the electric field intensity along the \( z \)-axis at a distance \( Z \) from origin:

Figure shows a rod \( AB \), which is bent in a \( 120^\circ \) circular arc of radius \( R \). A charge \( (-Q) \) is uniformly distributed over rod \( AB \). What is the electric field \( \vec{E} \) at the centre of curvature \( O \)?