A ball of radius $R$ carries a positive charge whose volume density depends only on the separation $r$ from the ball’s centre as $\rho ={\rho }_0 \left(\frac{1 – r}{R}\right)$, where ${\rho }_0$ is a constant. Assuming the permittivity of the ball and the environment to be equal to unity, find :

(i) The magnitude of the electric field strength as a function of the distance r both inside and outside the ball;

(ii) The maximum intensity $E_{\max }$ and the corresponding distance $r_{\mathrm{m}}$.

Consider a region in free space bounded by the surfaces of an imaginary cube having sides of length $a$ as shown in the figure. A charge $+Q$ is placed at the centre $O$ of the cube. $P$ is such a point outside the cube that the line $OP$perpendicularly intersects the surface $ABCD$ at $R$ and also $OR=RP=a/2$. A charge $+Q$ is placed at point $P$ also. What is the total electric flux through the five faces of the cube other than $ABCD?$

What is the electric flux for Gaussian surface A that encloses the charged particles in free space? [Given: $\left.q_1=-14 \mathrm{nC},q_2=78.85 \mathrm{nC},q_3=-56 \mathrm{nC}\right]$

Given below are two statements
Statement $\mathrm{I}$ : An electric dipole is placed at the centre of a hollow sphere. The flux of electric field through the sphere is zero, but the electric field is not zero anywhere in the sphere.

Statement $\mathrm{II}$ : If $R$ is the radius of a solid metallic sphere and $Q$ be the total charge on it. The electric field at any point on the spherical surface of radius $r\left(<R\right)$ is zero but the electric flux passing through this closed spherical surface of radius $r$ is not

In the light of the above statements, choose the correct answer from the options given below:

Arrangement of charges are shown in the figure. Flux linked with the closed surface $P$ and $Q$ respectively are ________ and ________ .

When a $10 \mathrm{\mu C}$ charge is enclosed by a closed surface, the flux passing through the surface is $\varphi$. Now another charge $-5\mathrm{\mu C}$ is placed inside the same closed surface, then the flux passing through the surface is _____.