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 $120^{\circ}$ 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 is (are) true?

$(A)$ The electric flux through the shell is $\sqrt{3} R \lambda / \epsilon_0$

$(B)$ The z-component of the electric field is zero at all the points on the surface of the shell

$(C)$ The electric flux through the shell is $\sqrt{2} R \lambda / \epsilon_0$

$(D)$ The electric field is normal to the surface of the shell at all points

Figure shows the electric lines of force emerging from a charged body. If the electric field at $A$ and $B$ are ${E_A}$ and ${E_B}$ respectively and if the displacement between $A$ and $B$ is $r$ then

A disk of radius $a / 4$ having a uniformly distributed charge $6 \mathrm{C}$ is placed in the $x-y$ plane with its centre at $(-a / 2,0,0)$. A rod of length $a$ carrying a uniformly distributed charge $8 \mathrm{C}$ is placed on the $x$-axis from $x=a / 4$ to $x=5 a / 4$. Two point charges $-7 \mathrm{C}$ and $3 \mathrm{C}$ are placed at $(a / 4,-a / 4,0)$ and $(-3 a / 4,3 a / 4,0)$, respectively. Consider a cubical surface formed by six surfaces $x= \pm a / 2, y= \pm a / 2$, $z= \pm a / 2$. The electric flux through this cubical surface is

Four closed surfaces and corresponding charge distributions are shown below

Let the respective electric fluxes through the surfaces be ${\phi _1},{\phi _2},{\phi _3}$ and ${\phi _4}$ . Then

A positive charge $q$ is kept at the center of a thick shell of inner radius $R_1$ and outer radius $R_2$ which is made up of conducting material. If $\phi_1$ is flux through closed gaussian surface $S_1$ whose radius is just less than $R_1$ and $\phi_2$ is flux through closed gaussian surface $S_2$ whose radius is just greater than $R_1$ then:-

The magnitude of the average electric field normally present in the atmosphere just above the surface of the Earth is about $150\, N/C$, directed inward towards the center of the Earth . This gives the total net surface charge carried by the Earth to be......$kC$ [Given ${\varepsilon _0} = 8.85 \times {10^{ - 12}}\,{C^2}/N - {m^2},{R_E} = 6.37 \times {10^6}\,m$]