Which of the following correctly represents the variation of electric potential $(V)$ of a charged spherical conductor of radius $(R)$ with radial distance $(r)$ from the centre?

A thin spherical insulating shell of radius $R$ carries a uniformly distributed charge such that the potential at its surface is $V _0$. A hole with a small area $\alpha 4 \pi R ^2(\alpha<<1)$ is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?

Consider a sphere of radius $R$ having charge $q$ uniformly distributed inside it. At what minimum distance from its surface the electric potential is half of the electric potential at its centre?

Consider two points $1$ and $2$ in a region outside a charged sphere. Two points are not very far away from the sphere. If $E$ and $V$ represent the electric field vector and the electric potential, which of the following is not possible

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 ($2$)

Charges are placed on the vertices of a square as shown Let $\vec E$ be the electric field and $V$ the potential at the centre. If the charges on $A$ and $B$ are interchanged with those on $D$ and $C$ respectively, then