An electric charge $10^{-6} \mu \mathrm{C}$ is placed at origin $(0,0)$ $\mathrm{m}$ of $\mathrm{X}-\mathrm{Y}$ co-ordinate system. Two points $\mathrm{P}$ and $\mathrm{Q}$ are situated at $(\sqrt{3}, \sqrt{3}) \mathrm{m}$ and $(\sqrt{6}, 0) \mathrm{m}$ respectively. The potential difference between the points $P$ and $Q$ will be :

Consider an evacuated cylindrical chamber of height $h$ having rigid conducting plates at the ends and an insulating curved surface as shown in the figure. A number of spherical balls made of a light weight and soft material and coated with a conducting material are placed on the bottom plate. The balls have a radius $r \ll h$. Now a high voltage source ($HV$) is connected across the conducting plates such that the bottom plate is at $+V_0$ and the top plate at $-V_0$. Due to their conducting surface, the balls will get charged, will become equipotential with the plate and are repelled by it. The balls will eventually collide with the top plate, where the coefficient of restitution can be taken to be zero due to the soft nature of the material of the balls. The electric field in the chamber can be considered to be that of a parallel plate capacitor. Assume that there are no collisions between the balls and the interaction between them is negligible. (Ignore gravity)

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($1$) Which one of the following statements is correct?

($A$) The balls will stick to the top plate and remain there

($B$) The balls will bounce back to the bottom plate carrying the same charge they went up with

($C$) The balls will bounce back to the bottom plate carrying the opposite charge they went up with

($D$) The balls will execute simple harmonic motion between the two plates

($2$) The average current in the steady state registered by the ammeter in the circuit will be

($A$) zero

($B$) proportional to the potential $V_0$

($C$) proportional to $V_0^{1 / 2}$

($D$) proportional to $V_0^2$

Give the answer quetion ($1$) and ($2$)

Two charge $ + \,q$ and $ - \,q$ are situated at a certain distance. At the point exactly midway between them

Three concentric spherical shells have radii $a, b$ and $c (a < b < c)$ and have surface charge densities $\sigma ,-\;\sigma $ and $\;\sigma \;$ respectively. If  $V_A,V_B$ and $V_C$  denote the potentials of the three shells, then, for $c = a +b,$ we have

In a region, if electric field is defined as $\vec E = \left( {\hat i + 2\hat j + \hat k} \right)\,V/m$ , then the potential difference between two points $A (0, 0, 0)$ and $B (2, 3, 4)$ in that region, is ......$V$

Two identical positive charges are placed at $x =\, -a$ and $x = a$ . The correct variation of potential $V$ along the $x-$ axis is given by