Two electrons are moving towards each other, each with a velocity of $10^6 \,m / s$. What will be closest distance of approach between them is ......... $m$

In space of horizontal $EF$ ($E = (mg)/q$) exist as shown in figure and a mass $m$ attached at the end of a light rod. If mass $m$ is released from the position shown in figure find the angular velocity of the rod when it passes through the bottom most position

An electron enters in high potential region ${V_2}$ from lower potential region ${V_1}$ then its velocity

This questions has statement$-1$ and statement$-2$. Of the four choices given after the statements, choose the one that best describe the two statements.
An insulating solid sphere of radius $R$ has a uniformly
positive charge density $\rho$. As a result of this uniform charge distribution there is a finite value of electric potential at the centre of the sphere, at the surface of the sphere and also at a point out side the sphere. The electric potential at infinite is zero.

Statement$ -1$ : When a charge $q$ is take from the centre of the surface of the sphere its potential energy changes by  $\frac{{q\rho }}{{3{\varepsilon _0}}}$

Statement$ -2$ : The electric field at a distance $r(r < R)$  from centre of the sphere is $\frac{{\rho r}}{{3{\varepsilon _0}}}$

A charged particle of charge $Q $ is held fixed and another charged particle of mass $m$ and charge $q$ (of the same sign) is released from a distance $r.$ The impulse of the force exerted by the external agent on the fixed charge by the time distance between $Q$ and $q$ becomes $2r$ is

A disk of radius $R$ with uniform positive charge density $\sigma$ is placed on the $x y$ plane with its center at the origin. The Coulomb potential along the $z$-axis is

$V(z)=\frac{\sigma}{2 \epsilon_0}\left(\sqrt{R^2+z^2}-z\right)$

A particle of positive charge $q$ is placed initially at rest at a point on the $z$ axis with $z=z_0$ and $z_0>0$. In addition to the Coulomb force, the particle experiences a vertical force $\vec{F}=-c \hat{k}$ with $c>0$. Let $\beta=\frac{2 c \epsilon_0}{q \sigma}$. Which of the following statement($s$) is(are) correct?

$(A)$ For $\beta=\frac{1}{4}$ and $z_0=\frac{25}{7} R$, the particle reaches the origin.

$(B)$ For $\beta=\frac{1}{4}$ and $z_0=\frac{3}{7} R$, the particle reaches the origin.

$(C)$ For $\beta=\frac{1}{4}$ and $z_0=\frac{R}{\sqrt{3}}$, the particle returns back to $z=z_0$.

$(D)$ For $\beta>1$ and $z_0>0$, the particle always reaches the origin.