The charge distribution along the semi-circular arc is non-uniform . Charge per unit length $\lambda $ is given as $\lambda  = {\lambda _0}\sin \theta $ , with $\theta $ measured as shown in figure. $\lambda_0$ is a positive constant. The radius of arc is $R$ . The electric field at the center $P$ of semi-circular arc is $E_1$ . The value of $\frac{{{\lambda _0}}}{{{ \in _0}{E_1}R}}$ is

An electron experiences a force equal to its weight when placed in an electric field. The intensity of the field will be

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 equal negative charges $-\, q$ each are fixed at the points $(0, a)$ and $(0, -a)$ on the $Y$ -axis. A positive charge $Q$ is released from rest at the point $(2a, 0)$ on the $X$ -axis. The charge $Q$ will :-

A positively charged ball hangs from a silk thread. We put a positive test charge ${q_0}$ at a point and measure $F/{q_0}$, then it can be predicted that the electric field strength $E$

Two point charges $Q$ and $-3Q$ are placed at some distance apart. If the electric field at the location of $Q$ is $E$ then at the locality of $ - 3Q$, it is