Select the correct statement$(s)$:
$(1)$ The density of electric field lines is independent of the magnitude of the electric field vector $E$ at a given point.
$(2)$ The density of electric field lines is proportional to the magnitude of the electric field vector $E$ at a given point.
$(3)$ Electric field lines do not exist in reality. They are only a graphical representation of the electric field.
$(4)$ Electric field lines exist in reality.

Consider the four surfaces $S_1, S_2, S_3,$ and $S_4$ enclosing the same charge $q_1$ as shown in the figure. Compare the electric flux through these surfaces.

Choose the incorrect statement:
$(a)$ The electric lines of force entering into a Gaussian surface provide negative flux.
$(b)$ $A$ charge '$q$' is placed at the centre of a cube. The flux through all the faces will be the same.
$(c)$ In a uniform electric field,the net flux through a closed Gaussian surface containing no net charge is zero.
$(d)$ When the electric field is parallel to a Gaussian surface,it provides a finite non-zero flux.
Choose the most appropriate answer from the options given below:

In a region,the intensity of an electric field is given by $E = 2i + 3j + k$ in $NC^{-1}$. The electric flux through a surface $S = 10i \ m^2$ in the region is

Gauss's law should be invalid if

Draw the electric field lines for a system of two identical positive charges placed near each other.