Two mutually perpendicular infinite wires along $x$-axis and $y$-axis carries charge densities, ${\lambda }_1$and ${\lambda }_2$. The electric line of force at $P$ is along the line $y=\frac{x}{\sqrt{3}}$ where $P$ is also a point lying on the same line. Find $\frac{{\lambda }_1}{{\lambda }_2}$.

Figure shows, in cross-section, two solid spheres with uniformly distributed charge throughout their volumes. Each has radius $R$. Point $P$ lies on a line connecting the centres of the spheres, at radial distance $\frac{R}{2}$ from the centre of sphere $1$. If the net electric field at point $P$ is zero, and $Q_1$ is $64 \mathrm{\mu C}$ and $Q_2=8x \mathrm{\mu C}$, what is the value of $x$?

It has been experimentally observed that the electric field in a large region of earth's atmosphere is directed vertically down. At an altitude of $300 \mathrm{m}$, the electric field is $60 \mathrm{V} {\mathrm{m}}^{-1}$ • At an altitude of $200 \mathrm{m}$, the field $100 \mathrm{V} {\mathrm{m}}^{-1}$. The net amount of charge contained in the cube of $100 \mathrm{m}$ Edge, located between $200 \mathrm{m}$ and $300 \mathrm{m}$. Altitude is found to be $n\times {10}^5{\epsilon }_0 C$. Find the value of $n$.

A point charge $Q$ is located on the axis of disc of radius $R$ at a distance $b$ from the plane of the disk. If one fourth of the electric flux from the charge passes through the disk, then $R=\sqrt{\square }b$ (as shown in figure). Suitable value for $\square$ can be,

${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ are two hollow concentric spheres with charge $q$ and $2q.$ Space between ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ is filled with a dielectric of dielectric constant, $K=5$. The ratio of flux through $S_2$ and flux through $S_1$ is $K\text{'}.$ Then find the value of $15K\text{'}.$