• Distance between the two point charges along the $x$-axis:
  $$r = 6\,\text{cm} - (-6\,\text{cm}) = 12\,\text{cm} = 0.12\,\text{m}$$
• The initial electrostatic potential energy $U_i$ of the two-charge system is:
  $$U_i = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r} = \frac{(9 \times 10^9)(4 \times 10^{-6})(-3 \times 10^{-6})}{0.12} = \frac{-0.108}{0.12} = -0.9\,\text{J}$$
• At infinite separation, the final potential energy is $U_\infty = 0$.
• The work required by an external agent to separate the charges infinitely away without acceleration is:
  $$W = U_\infty - U_i = 0 - (-0.9\,\text{J}) = +0.9\,\text{J}$$

The electrostatic potential at distance $r$ from an electric dipole with dipole moment $\vec{p} = q(2\vec{a})$ is:

• $V$ explicitly depends on the radial distance $r$, making statement (A) correct.
• $V$ depends on the orientation angle $\theta$ between position vector $\vec{r}$ and dipole vector $\vec{p}$, making statement (B) correct.
• At distances large compared to dipole size ($r \gg a$), the potential falls off as $1/r^2$, making statement (C) correct.
• Since dipole moment $p = q(2a)$ depends directly on the separation $2a$ between charges, the potential does depend on the distance separating the charges. Hence (D) is incorrect.

When two conducting spheres are brought into contact, charge flows between them until they reach a common electrical potential $V$ ($V_1 = V_2 = V$).
The electric potential of an isolated spherical conductor carrying charge $Q$ and radius $R$ is:

Equating potentials:

Analysis of statements:

• Statement A is correct: By the law of conservation of charge, the net charge of an isolated system remains constant.
• Statement B is incorrect: In a conductor, excess charge redistributes over the outer surface to minimize electrostatic potential energy.
• Statement C is correct: $q = ne \implies n = \frac{1}{1.6 \times 10^{-19}} = 6.25 \times 10^{18}$ electrons.
• Statement D is incorrect: Electric field is a vector field (it has both magnitude and direction).
• Statement E is correct: A permanent dipole has a permanent intrinsic electric dipole moment even in the absence of an external electric field (e.g., water molecules).