The linear charge density on a dielectric ring of radius $R$ varies with $\theta $ as $\lambda \, = \,{\lambda _0}\,\cos \,\,\theta /2,$ where $\lambda _0$ is constant. Find the potential at the centre $O$ of ring. [in volt]

Two point charges $-Q$ and $+Q / \sqrt{3}$ are placed in the xy-plane at the origin $(0,0)$ and a point $(2,0)$, respectively, as shown in the figure. This results in an equipotential circle of radius $R$ and potential $V =0$ in the $xy$-plane with its center at $(b, 0)$. All lengths are measured in meters.

($1$) The value of $R$ is. . . . meter.

($2$) The value of $b$ is. . . . . .meter.

Three concentric spherical metallic shells $X , Y$ and $Z$ of radius $a , b$ and c respectively $[ a < b < c ]$ have surface charge densities $\sigma,-\sigma$ and $\sigma$, respectively. The shells $X$ and $Z$ are at same potential. If the radii of $X$ and $Y$ are $2\,cm$ and $3\,cm$, respectively.The radius of shell $Z$ is $......cm$.

Figure shows the variation of electric field intensity $E$ versus distance $x$. What is the potential difference between the points at $x=2 \,m$ and at $x=6 \,m$ from $O$ is ............. $V$

Six point charges are kept at the vertices of a regular hexagon of side $L$ and centre $O$, as shown in the figure. Given that $K=\frac{1}{4 \pi \varepsilon_0} \frac{q}{L^2}$, which of the following statement $(s)$ is (are) correct?

$(A)$ the elecric field at $O$ is $6 K$ along $O D$

$(B)$ The potential at $O$ is zero

$(C)$ The potential at all points on the line $PR$ is same

$(D)$ The potential at all points on the line $ST$ is same.

Calculate potential on the axis of a ring due to charge $Q$ uniformly distributed along the ring of radius $R$.