A uniform electric field, $$\overrightarrow E = - 400\sqrt 3 \widehat y$$ NC⁻¹ is applied in a region. A charged particle of mass m carrying positive charge q is projected in this region with an initial speed of 2$$\sqrt {10} $$ $$ \times $$ 10⁶ ms⁻¹ . This particle is aimed to hit a target T, which is 5 m away from its entry point into the field as shown schematically in the figure.
Take $${q \over m}$$ = 10¹⁰ Ckg⁻¹ . Then

An electric dipole with dipole moment $${{{p_0}} \over {\sqrt 2 }}(\widehat i + \widehat j)$$ is held fixed at the origin O in the presence of a uniform electric field of magnitude E₀.



If the potential is constant on a circle of radius R centered at the origin as shown in figure, then the correct statement(s) is/are, ($$ \in $$₀ is the permittivity of the free space, R >> dipole size)

A charged shell of radius R carries a total charge Q. Given $$\phi $$ as the flux of electric field through a closed cylindrical surface of height h, radius r and with its center same as that of the shell. Here, center of the cylinder is a point on the axis of the cylinder which is equidistant from its top and bottom surfaces. Which of the following option(s) is/are correct?

[$$ \in $$₀ is the permittivity of free space]

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 center $$O$$ of the spherical shell, as shown in the figure. The permittivity of free space is $${ \in _0}.$$ Which of the following statement is (are) true?