Gauss’s Law and Its Applications

A frustum is cut from a right circular cone. The two circular faces have radii $R$and $2R$ and their centers are at $O_1$ and $O_2$ respectively. Height of the frustum is $h=3R$. When a point charge $Q$ is placed at $O_1$, the flux of electric field through the circular face of radius $2R$ is ${\varphi }_1$and when the charge $Q$ is placed at $O_2$ , the flux through the other circular face is ${\varphi }_2.$The ratio of $\left(\frac{{\varphi }_1}{{\varphi }_2}\right)$ is found to be $\left(\frac{\sqrt{13}-3}{\sqrt{10}-3}\right)\left(\sqrt{\frac{a}{b}}\right)$. The value of $(a+b)$ is

A circular disc of radius $\mathrm{R}$ carries surface charge density $\sigma \left(\mathrm{r}\right)={\sigma }_0\left(1-\frac{\mathrm{r}}{\mathrm{R}}\right),$ where ${\sigma }_0$ is a constant and $\mathrm{r}$ 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 $\frac{\mathrm{R}}{4}$ and concentric with the disc is $\varphi .$ Then the ratio is $\frac{{\varphi }_0}{\varphi }=\frac{x}{5}$, the value of $x$ is___

The electric field inside a sphere which carries a volume charge density proportional to the distance from the origin $\rho =\alpha r$ ($\alpha$ is a constant) is:

A very long charged solid cylinder of radius $a$ contains a uniform charge density $\rho$. Dielectric constant of the material of the cylinder is $K .$ What will be the magnitude of electric field at a radial distance $x\left(x<a\right)$ from the axis of the cylinder ?

An infinitely long thin non-conducting wire is parallel to the $z-$axis and carries a uniform line charge density $\mathrm{\lambda }$. It pierces a thin non-conducting spherical shell of radius $R$ in such a way that the arc $\mathrm{PQ}$  subtends an angle ${120}^{°}$ at the centre O of the spherical shell, as shown in the figure. The permittivity of free space is ${\mathrm{\epsilon }}_0$. Which of the following statements is (are) true? 

A point charge $+Q$ is placed just outside an imaginary hemispherical surface of radius $R$ as shown in the figure. Which of the following statements is/are correct? 

Two non-conducting spheres of radii $R_1$ and $R_2$ and carrying uniform volume charge densities $+\mathrm{\rho }$ and $-\mathrm{\rho }$, respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region

A solid sphere of the radius $r$ has a charge $Q$ distributed in its volume with a charge density $\rho =k{r}^{\mathrm{a}}$ where $k$ and $a$ are constants, and $r$ is the distance from the centre. If the electric field at $r=R/2$ is $\frac{1}{8}$ times that at $r=R$, Find the value of $a$.

An infinitely long thin non-conducting wire is parallel to the $z$-axis and carries a uniform line charge density $\lambda$. It pierces a thin non-conducting spherical shell of radius $R$ in such a way that the arc $PQ$ subtends an angle $120°$ at the centre $O$ of the spherical shell, as shown in the figure. The permittivity of free space is ${\epsilon }_0$. Which of the following statements (are) true?

Two uniformly long charged wires with linear densities $\text{λ}$ and $3\text{λ}$ are placed along $\mathrm{X}$ and $\mathrm{Y}$ axis respectively. Determined the slope of electric field at any point on the line $\mathrm{y}=\sqrt{3}\mathrm{x}$ .