An electric dipole of dipole moment $\vec{p}$ is placed at the origin of the coordinate system along the $z$-axis. The amount of work required to move a charge $q$ from the point $A(a, 0, 0)$ to the point $B(0, 0, a)$ is,

The work done in rotating a dipole placed parallel to the electric field through $180^{\circ}$ is $w$. What is the work done in rotating it through $60^{\circ}$? $(\cos 0^{\circ}=1, \cos 60^{\circ}=\frac{1}{2}, \cos 180^{\circ}=-1)$

Two charges $-q$ each are separated by a distance $2d$. $A$ third charge $+q$ is kept at the midpoint $O$. Find the potential energy of $+q$ as a function of a small distance $x$ from $O$ due to the $-q$ charges. Sketch the $P.E.$ versus $x$ graph and convince yourself that the charge at $O$ is in an unstable equilibrium.

Three point charges shown in the figure lie along a straight line. The energy required to exchange the position of the central charge with one of the negative charges is

If one of the two electrons of a $H_{2}$ molecule is removed, we get a hydrogen molecular ion $H_{2}^{+}$. In the ground state of an $H_{2}^{+}$, the two protons are separated by roughly $1.5 \; Å,$ and the electron is roughly $1 \; Å$ from each proton. Determine the potential energy of the system. Specify your choice of the zero of potential energy. (in $; eV.$)

An electric dipole with dipole moment $\vec p$ is placed in a uniform electric field $\vec E$. The work done in rotating the dipole by $90^\circ$ from its initial equilibrium position is: