Embibe Experts Solutions for Chapter: Electrostatics, Exercise 1: JEE Advanced Paper 1 - 2013

Two identical non-conducting solid spheres of same mass and charge are suspended in air from a common point by two non-conducting, massless strings of same length. At equilibrium, the angle between the strings is $\alpha$. The spheres are now immersed in a dielectric liquid of density $800 \mathrm{kg} {\mathrm{m}}^{-3}$ and dielectric constant $21$. If the angle between the strings remains the same after the immersion, then

Two capacitors with capacitance values $C_1=\left(2000\pm 10\right) \mathrm{pF}$ and $C_2=\left(3000\pm 15\right) \mathrm{pF}$ are connected in series. The voltage applied across this combination is $V=\left(5.00\pm 0.02\right) \mathrm{V}$. The percentage error in the calculation of the energy stored in this combination of capacitors is __________.

An electric dipole with dipole moment $\frac{{\mathrm{p}}_0}{\sqrt{2}}\left(\widehat{\mathrm{i}}+\widehat{\mathrm{j}}\right)$ is held fixed at the origin $\mathrm{O}$ in the presence of an uniform electric field of magnitude ${\mathrm{E}}_0.$ If the potential is constant on a circle of radius $\mathrm{R}$ centered at the origin as shown in figure, then the correct statement(s) is/are:
( ${\mathrm{\epsilon }}_0$ is permittivity of free space, $\mathrm{R}\gg$ dipole size)

A uniform electric field, $\overrightarrow{E}=-400\sqrt{3} \hat{\mathrm{j}} \mathrm{N} {\mathrm{C}}^{-1}$ is applied in a region. A charged particle of mass $m$ carrying positive charge $q$ is projected in this region with an initial speed of $2\sqrt{10}\times {10}^6 \mathrm{m} {\mathrm{s}}^{-1}$. This particle is aimed to hit a target $T$, which is $5 \mathrm{m}$ away from its entry point into the field as shown schematically in the figure. Take $\frac{q}{m}={10}^{10} \mathrm{C} {\mathrm{kg}}^{-1}$. Then

A circular disc of radius $R$ carries surface charge density $\sigma \left(r\right)={\sigma }_0\left(1-\frac{r}{R}\right)$, where ${\sigma }_0$ is a constant and $r$ is the distance from the center of the disc. Electric flux through a large spherical surface that encloses the charged disc completely is ${\varphi }_0$. Electric flux through another spherical surface of radius $\frac{R}{4}$ and concentric with the disc is $\varphi$. Then the ratio $\frac{{\varphi }_0}{\varphi }$ is

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

A charged shell of radius $\mathrm{R}$ carries a total charge $\mathrm{Q}.$ Given $\mathrm{\Phi }$ as the flux of electric field through a closed cylindrical surface of height $\mathrm{h},$ radius $\mathrm{r}$ and with its center same as that of the shell. Here, center of the cylinder is a point on the axis of the cylinder which is equidistant from its top and bottom surfaces. Which of the following option(s) is/are correct? [ ${\in }_0$ is the permittivity of free space]

A parallel plate capacitor of capacitance $\mathrm{C}$ has spacing $d$ between two plates having area $\mathrm{A}.$ The region between the plates is filled with $\mathrm{N}$ dielectric layers, parallel to its plates, each with thickness $\mathrm{\delta }=\frac{\mathrm{d}}{\mathrm{N}}.$ The dielectric constant of the ${\mathrm{m}}^{\mathrm{t}\mathrm{h}}$ layer is ${\mathrm{K}}_{\mathrm{m}}=\mathrm{K}\left(1+\frac{\mathrm{m}}{\mathrm{N}}\right).$ For a very large $\mathrm{N}\left(>{10}^3\right),$ the capacitance $\mathrm{C}$ is $\mathrm{\alpha }\left(\frac{\mathrm{K}{\in }_0\mathrm{A}}{\mathrm{d}\mathrm{l}\mathrm{n}2}\right).$ The value of $\mathrm{\alpha }$ will be____.

$[{\in }_0$ is the permittivity of free space]