Two small balls of mass $m$ each with charges $+q$ and $-q$ respectively, connected by a rigid massless rod of length $d$, form a dipole. The dipole is parallel to plane $X Y$ and is placed in a uniform magnetic field $\vec{B}$ perpendicular to $X Y$.

Initially, the dipole is aligned with the direction $X$ and has initial angular velocity $\omega_{0}$ in plane $X Y$, as shown. Its center of mass is initially located at origin and given initial velocity $\vec{v}_{0}$ parallel to $X Y$, as well.

Consider three distinct scenarios (a, b, c-d):

(a) Find $\omega_{0}$ and the direction of $\vec{v}_{0}$, so that the center of mass will move with the constant velocity $\vec{v}=\vec{v}_{0}$ ?

A S

(b) Given $\omega_{0}$, find such $\vec{v}_{0}$ (direction and magnitude), so that the center of mass will travel in a circle. Find the circle radius $R_{c}$ and the coordinates $x_{c}, y_{c}$ of its center. You don't need to prove the uniqueness of the solution.

A S

(c) Given $\vec{v}_{0}=0$, find the minimal $\omega_{0}=\omega_{\text {min }}$ necessary for the dipole to reverse its orientation during the motion.

A S

(d) If the dipole starts with $\vec{v}_{0}=0$ and $\omega_{0}=\omega_{\text {min }}$ found in part (c), the trajectory of its center of mass has an asymptote. Find the distance $D$ from the origin to the asymptote.

A S

Useful vector identity:

$$ \vec{a} \times(\vec{b} \times \vec{c})=\vec{b}(\vec{a} \cdot \vec{c})-\vec{c}(\vec{a} \cdot \vec{b}) $$ where " $\times$ " and "$\cdot$" denote vector product and scalar product respectively.