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.

Concentric metallic hollow spheres of radii $R$ and $4 R$ hold charges $Q _{1}$ and $Q _{2}$ respectively. Given that surface charge densities of the concentric spheres are equal, the potential difference $V ( R )- V (4 R )$ is

Prove that, if an insulated, uncharged conductor is placed near a charged conductor and no other conductors are present, the uncharged body must be intermediate in potential between that of the charged body and that of infinity.

A charge $+q$ is fixed at each of the points $x = x_0,\,x = 3x_0,\,x = 5x_0$, .... upto $\infty $ on $X-$  axis and charge $-q$ is fixed on each of the points $x = 2x_0,\,x = 4x_0,\,x = 6x_0$, .... upto $\infty $ . Here $x_0$ is a positive constant. Take the potential at a point due to a charge $Q$ at a distance $r$ from it to be $\frac{Q}{{4\pi {\varepsilon _0}r}}$. Then the potential at the origin due to above system of charges will be

Considering a group of positive charges, which of the following statements is correct?

Two thin wire rings each having a radius $R$ are placed at a distance $d$ apart with their axes coinciding. The charges on the two rings are $ + q$ and $ - q$. The potential difference between the centres of the two rings is