The electric flux for Gaussian surface A that enclose the charged particles in free space is (given $q_1$ = $-14\, nC$, $q_2$ = $78.85\, nC$, $q_3$ = $-56 \,nC$)

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

A point charge causes an electric flux of $-1.0 \times 10^{3}\; N\;m ^{2} / C$ to pass through a spherical Gaussian surface of $10.0\; cm$ radius centred on the charge.

$(a)$ If the radius of the Gaussian surface were doubled, how much flux would pass through the surface?

$(b)$ What is the value of the point charge?

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 rectangular surface of sides $10 \,cm$ and $15 \,cm$ is placed inside acyniform electric field of $25 \,V / m$, such that the surface makes an angle of $30^{\circ}$ with the direction of electric field. Find the flux of the electric field through the rectangular surface .................. $Nm ^2 / C$

$(a)$ An electrostatic field line is a continuous curve. That is, a field line cannot have sudden breaks. Why not?

$(b)$ Explain why two field lines never cross each other at any point?