The electric field in a certain region is acting radially outward and is given by $E = A r.$ A charge contained in a sphere of radius $a$ centred at the origin of the field, will be given by

Two identical charged spheres suspended from a common point by two massless strings of lengths l, are initially at a distance d (d << l) apart because of their mutual repulsion. The charges begin to leak from both the spheres at a constant rate. As a result, the spheres approach each other with a velocity v. The v varies as a function of the distance x between the spheres, as

Suppose the charge of a proton and an electron differ slightly. One of them is $-e,$ the other is $(e + \Delta e).$ If the net of electrostatic force and gravitational force between two hydrogen atoms placed at a distance $d$ (much greater than atomic size) apart is zero, then $\Delta e$ is of the order of (Given: mass of hydrogen $m_h = 1.67 \times 10^{-27}$ kg)

An electron falls from rest through a vertical distance $h$ in a uniform and vertically upward directed electric field $E.$ The direction of the electric field is now reversed, keeping its magnitude the same. A proton is allowed to fall from rest in it through the same vertical distance $h.$ The time of fall of the electron, in comparison to the time of fall of the proton is

A hollow metal sphere of radius $R$ is uniformly charged. The electric field due to the sphere at a distance $r$ from the center

Two parallel infinite line charges with linear charge densities $+\lambda$ C/m and $-\lambda$ C/m are placed at a distance of $2R$ in free space. What is the electric field mid-way between the two line charges?

Two point charges A and B, having charges $+Q$ and $-Q$ respectively, are placed at certain distance apart and force acting between them is $F.$ If 25% charge of A is transferred to B, then force between the charges becomes:

A short electric dipole has a dipole moment of $16 \times 10^{-9}$ C m. The electric potential due to the dipole at a point at a distance of $0.6$ m from the centre of the dipole, situated on a line making an angle of $60^{\circ}$ with the dipole axis is:$\newline \left( \frac{1}{4\pi\epsilon_0} = 9 \times 10^9 { N m}^2 {C}^{-2} \right)$

A spherical conductor of radius $10$ cm has a charge of $3.2 \times 10^{-7}$ C distributed uniformly. What is the magnitude of the electric field at a point $15$ cm from the center of the sphere?$\newline (\frac{1}{4\pi\varepsilon_0} = 9 \times 10^9 { N m}^2{ C}^{-2})$

Two charged spherical conductors of radius $R_1$ and $R_2$ are connected by a wire. Then the ratio of surface charge densities of the spheres $\left(\frac{\sigma_1}{\sigma_2}\right)$ is:

Two point charges $-q$ and $+q$ are placed at a distance of $L,$ as shown in the figure. The magnitude of electric field intensity at a distance $R$ ($R >> L$) varies as:

If $\oint \mathbf{E} \cdot d\mathbf{S} = 0$ over a surface, then:

An electric dipole is placed at an angle of $30^\circ$ with an electric field of intensity $2 \times 10^5 NC^{-1}.$ It experiences a torque equal to 4 Nm. Calculate the magnitude of charge on the dipole, if the dipole length is 2 cm.

An electric dipole is placed as shown in the figure. The electric potential (in $10^2 V$) at point P due to the dipole is ($\varepsilon_0$ = permittivity of free space and $K = \frac{1}{4 \pi \varepsilon_0}$):