Embibe Experts Solutions for Chapter: Electrostatics, Exercise 1: KVPY Aptitude Test - Stream SX (19 Nov 2017)

A proton and an antiproton come close to each other in vacuum such that the distance between them is $10 \mathrm{cm}.$ Consider the potential energy to be zero at infinity. The velocity at this distance will be:

Consider a sphere of radius $R$ with uniform charged density and total charge $Q.$ The electrostatic potential distribution inside the sphere is given by
$\varphi \left(r\right)=\frac{Q}{4\pi {\epsilon }_{0}R}\left[a+b{\left(\frac{r}{R}\right)}^c\right].$ Note that the zero of potential is at infinity. The values of $(a, b, c)$ are:

A charge $+q$ is distributed over a thin ring of radius $r$ with line charge density $\lambda =\frac{q{\sin }^2\theta }{\pi r}$. Note that the ring is in the $XY$-plane and $\theta$ is the angle made by $r$ with the $X$-axis. The work done by the electric force in displacing a point charge $+Q$ from the centre of the ring to infinity is

An electrostatic field line leaves at an angle $\alpha$ from point charge $q_1$ and connects with point charge $-q_2$ at an angle $\beta$($q_1$and $q_2$ are positive) see figure below. If $q_2=\frac{3}{2}{q}_1$ and $\alpha =30°,$ then

Two mutually perpendicular infinitely long straight conductors carrying uniformly distributed charges of linear densities ${\lambda }_1$ and ${\lambda }_2$ are positioned at a distance $r$ from each other.

Force between the conductors depends on $r$ as

Let the electrostatic field, $E$ at distance, $r$ from a point charge, $q$ not be an inverse square but instead an inverse cubic, e.g. $E=k\frac{q}{r^3}\hat{r}$, here $k$ is a constant. Consider the following two statements: (I) Flux through a spherical surface enclosing the charge is, $\varphi =\frac{q_{\mathrm{enclosed}}}{{\epsilon }_0}$. (II) A charge placed inside a uniformly charged shell will experience a force. Which of the above statements are valid?

The black shapes in the figure below are closed surfaces. The electric field lines are in red. For which case, the net flux through the surfaces is non-zero?

The potential due to an electrostatic charge distribution is $V\left(r\right)=\frac{q{e}^{-\alpha r}}{4\pi {\epsilon }_{0}r},$ where $\alpha$ is positive. The net charge within a sphere centred at the origin and of radius $1/\alpha$ is