An oil drop of $12$ excess electrons is held stationary under a constant electric field of $2.55 \times 10^{4}\; N\,C ^{-1}$ (Millikan's oil drop experiment). The density of the oil is $1.26 \;g \,cm ^{-3} .$ Estimate the radius of the drop. $\left(g=9.81\; m s ^{-2} ; e=1.60 \times 10^{-19}\; \,C \right)$

The intensity of the electric field required to keep a water drop of radius ${10^{ – 5}}\, cm$ just suspended in air when charged with one electron is approximately

Two identical non-conducting solid spheres of same mass and charge are suspended in air from a common point by two non-conducting, massless strings of same length. At equilibrium, the angle between the strings is $\alpha$. The spheres are now immersed in a dielectric liquid of density $800 kg m ^{-3}$ and dielectric constant $21$ . If the angle between the strings remains the same after the immersion, then

$(A)$ electric force between the spheres remains unchanged

$(B)$ electric force between the spheres reduces

$(C)$ mass density of the spheres is $840 kg m ^{-3}$

$(D)$ the tension in the strings holding the spheres remains unchanged

Two point charges $q_1$ and $q_2 (=q_1/2)$ are placed at points $A(0, 1)$ and $B(1, 0)$ as shown in the figure. The electric field vector at point $P(1, 1)$ makes an angle $\theta $ with the $x-$ axis, then the angle $\theta$ is

The unit of intensity of electric field is