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Let $P\left(r\right)=\frac{Q}{\pi R^4}r$ be the charge density distribution for a solid sphere of radius $R$ and total charge $Q.$ For a point $p$ inside the sphere at distance $r_1$ from the centre of the sphere, the magnitude of electric field is

Let there be a spherically symmetric charge distribution with charge density varying as $\rho \left(r\right)={\rho }_0\left(\frac{5}{4}-\frac{r}{R}\right)$ up to $r=R$, and $\rho \left(r\right)=0$ for $r>R$, where $r$ is the distance from the origin. The electric field at a distance $r(r<R)$ from the origin is given by

In a uniformly charged sphere of total charge $Q$ and radius $R$, the electric field $E$ is plotted as function of distance from the centre. The graph which would correspond to the above will be

This questions has Statement $1$ and Statement $2.$ Of the four choices given after the statements, choose the one that best describes the two statements.

An insulating solid sphere of radius $R$ has a uniformly positive charge density $\rho .$ As a result of this uniform charge distribution there is a finite value of the electric potential at the centre of the sphere, at the surface of the sphere and also at a point outside the sphere. The electric potential at infinity is zero.

Statement $1$: When a charge is taken from the centre to the surface of the sphere its potential energy changes by $\frac{q\rho }{3{\epsilon }_0}$
Statement $2$: The electric field at a distance $r(r<R)$ from the centre of the sphere is $\frac{\rho r}{3{\epsilon }_0}$

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?

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 solid cylinder of the radius $R$ has a uniform volume charge density $\rho$. It has a spherical cavity of the radius $\frac{R}{2}$ with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the electric field at the point $P$, which is at a distance $2R$ from the axis of the cylinder, is given by the expression $\frac{23\rho R}{16k{\epsilon }_0}$. The value of $k$ is 

An infinitely long uniform line charge distribution of charge per unit length $\lambda$ lies parallel to the $y$-axis in the $y-z$ plane at $z =\frac{\sqrt{3}}{2}a$ (see figure). If the magnitude of the flux of the electric field through the rectangular surface $ABCD$ lying in the $x-y$ plane with the centre at the origin is $\frac{\lambda l}{n{\epsilon }_0}$ (${\epsilon }_0$ is permittivity of free space ), then the value of $n$ is ____.