Two identical small balls each have a mass $m$and charge $q.$ When placed in a hemispherical bowl of radius $R$with frictionless, nonconductive walls, the beads move, and at equilibrium the line joining the balls is horizontal and the distance between them is $R$(see figure). Neglect any induced charge on the hemispherical bowl. Then the charge on each bead is

$\left(\text{ here k}=\frac{1}{4\pi {\epsilon }_0}\right)$

 

A thin semi-circular ring of radius $r$ has a positive charge $q$ distributed uniformly over it. The net field $E$ at the centre $O$ is

A solid cylindrical insulator of uniform density having length $4 \mathrm{m}$ and radius $2 \mathrm{m}$ contains charge $Q$. Find the value of the electric field at a distance $L$ along the axis from one end. ($L=\text{length of the insulator}$)

A block of mass $m$ is suspended vertically with a spring of spring constant $k.$ The block is made to oscillate in a gravitation field. Its time period is found to be $T.$ Now the space between the plates is made gravity free and an electric field $E$ is produced in the downward direction. Now the block is given a charge $q.$ The new time period of oscillation is, 

A thin glass rod is bent into a semicircle of radius $r$. A charge $+Q$ is uniformly distributed along the upper half, and a charge $-Q$ is uniformly distributed along the lower half, as shown in figure. The electric field $E$ at $P$, the centre of the semicircle, is

Find the electric field vector at $P$ $\left(a,a,a\right)$ due to three infinitely long lines of charges along the $x$-, $y$- and $z$ – axes, respectively. The charge density, i.e., charge per unit length of each wire is $\lambda$

Two semicircular rings lying in same plane, of uniform linear charge density $\lambda$ have radius $r$ and $2r$. They are joined using two straight uniformly charged wires of linear charge density $\lambda$ and length $r$ as shown in figure. The magnitude of electric field at common centre of semi circular rings is