A particle $A$ has charge $+q$ and particle $B$ has charge $+4 q$ with each of them having the same mass $m$. When allowed to fall from rest through the same electric potential difference, the ratio of their speeds $\frac{V_A}{V_B}$ will become

For an infinite line of charge having charge density $\lambda $ lying along $x-$ axis, the work required in moving charge $q$ from $C$ to $A$ along arc $CA$ is :-

The displacement of a charge $Q$ in the electric field $E = {e_1}\hat i + {e_2}\hat j + {e_3}\hat k$ is $\hat r = a\hat i + b\hat j$. The work done is

The ratio of momenta of an electron and an $\alpha$-particle which are accelerated from rest by a potential difference of $100\, volts$ is

In a hydrogen atom, the electron and proton are bound at a distance of about $0.53\; \mathring A:$

$(a)$ Estimate the potential energy of the system in $eV$, taking the zero of the potential energy at infinite separation of the electron from proton.

$(b)$ What is the minimum work required to free the electron, given that its kinetic energy in the orbit is half the magnitude of potential energy obtained in $(a)?$

$(c)$ What are the answers to $(a)$ and $(b)$ above if the zero of potential energy is taken at $1.06\;\mathring A$ separation?

A unit positive point charge of mass $m$ is projected with a velocity $V$ inside the tunnel as shown. The tunnel has been made inside a uniformly charged non conducting sphere. The minimum velocity with which the point charge should be projected such it can it reach the opposite end of the tunnel, is equal to