An electron of mass $m$ and charge $e$ is accelerated from rest through a potential difference $V$ in vacuum. The final speed of the electron will be

Two charges $-q$ each are fixed and separated by a distance $2d$. $A$ third charge $q$ of mass $m$ placed at the midpoint is displaced slightly by $x$ $(x \ll d)$ perpendicular to the line joining the two fixed charges as shown in the figure. Show that the charge $q$ will perform simple harmonic motion with a time period $T = \left[\frac{8 \pi^{3} \epsilon_{0} m d^{3}}{q^{2}}\right]^{1 / 2}$.

The intensity of the electric field required to keep a water drop of radius $10^{-5} \, cm$ and carrying a charge of $1$ electron suspended in air is:

An electron is made to enter symmetrically between two parallel and equally but oppositely charged metal plates,each of $10 \ cm$ length. The electron emerges out of the field region with a horizontal component of velocity $10^6 \ m/s$. If the magnitude of the electric field between the plates is $9.1 \ V/cm$,then the vertical component of velocity of the electron is (mass of electron $= 9.1 \times 10^{-31} \ kg$ and charge of electron $= 1.6 \times 10^{-19} \ C$)

$A$ positively charged particle moving along the $x$-axis with a certain velocity enters a uniform electric field directed along the positive $y$-axis. Its

An $\alpha$-particle of mass $6.4 \times 10^{-27} \ kg$ and charge $3.2 \times 10^{-19} \ C$ is situated in a uniform electric field of $1.6 \times 10^{5} \ Vm^{-1}$. The velocity of the particle at the end of $2 \times 10^{-2} \ m$ path when it starts from rest is