An infinitely long uniform line charge distribution of charge per unit length $\lambda$ lies parallel to the $y$-axis in the $y-z$ plane at $z =\frac{\sqrt{3}}{2}a$ (see figure). If the magnitude of the flux of the electric field through the rectangular surface $ABCD$ lying in the $x-y$ plane with the centre at the origin is $\frac{\lambda l}{n{\epsilon }_0}$ (${\epsilon }_0$ is permittivity of free space ), then the value of $n$ is ____.

A non-conducting ring of mass $m$ and radius $R$ is charged as shown. The linear charge density for corresponding region shown is $\lambda$ It is placed on tough horizontal surface. A uniform electric field $E_0\hat{i}$ is switched on and the ring starts rolling without sliding. Angular velocity acquired by the ring when it turns through an angle of $\frac{\pi }{2}$ is $16 \mathrm{rad} {\mathrm{s}}^{-1}.$ Value of $\frac{\lambda E_0}{m}$ (in $SI$ unit) is

A thin spherical shell of radius $R$ carries uniform surface charge density $\sigma .$ It is cut into two parts by a plane at a distance $\frac{R}{\sqrt{2}}$ from centre as shown. To hold two parts together force $F$ from both side has to be applied. The minimum force $F$ is $\frac{3\pi {\sigma }^2{R}^2}{n{\epsilon }_0}.$ Value of $n,$ is

An infinitely large layer of charge of uniform thickness $t$ is placed in existing uniform electric field. Presence of this charged layer so alters the electric field that it remains uniform on both the sides and assumes values $E_1$ and $E_2$ as shown in figure. If, $\left|{\overrightarrow{E}}_2\right|=20 \mathrm{N} {\mathrm{C}}^{-1}$ Then component of external field in $x$-direction (in $\mathrm{N} {\mathrm{C}}^{-1}$) is