B M Sharma Solutions for Chapter: Electric Flux and Gauss's Law, Exercise 2: DPP

A Gaussian surface $S$ encloses two charges $q_1=q$ and $q_2=-q$. The field at $P$ is, 

where ${\overrightarrow{E}}_1$, ${\overrightarrow{E}}_2$ and $\overrightarrow{E_3}$ are the field contributed by $q_1$, $q_2$ and $q_3$ at $P$ respectively.

A charge $q$ is placed at the centre of the open end of cylindrical vessel. The flux of the electric field through the surface of the vessel is, 

Electric charge is uniformly distributed along a long straight wire of radius $1 \mathrm{mm}$. The charge per $\mathrm{cm}$ length of the wire is $Q$ coulomb. Another cylindrical surface of radius $50 \mathrm{cm}$ and length $1 \mathrm{m}$ symmetrically encloses the wire as shown in the figure. The total electric flux passing through the cylindrical surface is, 

In a region of space having a spherically symmetric distribution of charge, the electric flux enclosed by a concentric spherical Gaussian surface varies with a radius $r$ as 
$\mathrm{\varphi }=\left\{\begin{array}{ll}\frac{{\mathrm{\varphi }}_0{r}^2}{R^3},& r\le R\\ {\mathrm{\varphi }}_0,& r>R\end{array}\right.$ where $R$ and ${\mathrm{\varphi }}_0$ are the constants.

The electric field strength in the region is given as, 

In a region of space, the electric field is in the $x$-direction and proportional to $x$, i.e., $\overrightarrow{\mathrm{E}}=E_{0}x\hat{\mathrm{i}}$. Consider an imaginary cubical volume of edge $a$ with its edges parallel to the axes of coordinates. The charge inside this volume will be, 

The figure shows a closed surface that intersects a conducting sphere. If a positive charge is placed at the point $P$, the flux of the electric field through the closed surface, 

The figure shows a neutral metallic sphere with a point charge $+Q$ placed near its surface. Electrostatic equilibrium conditions exist on metallic sphere. Mark the correct statements.

$A$ and $B$ are semi-spherical surfaces of radius $r_1$ and $r_2$ $\left(r_1>r_2\right)$ with $E_1$ and $E_2$ as the electric fields at their surfaces. Charge $q_0$ is placed as shown. What is the condition which may be satisfied?