(d) (1 pt) If the dipole starts with and found in part (c), the trajectory... · OlympiadBench #1509

Physics · electromagnetism

Problem

(d) (1 pt) 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. Given constants: 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

. <image_1> 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): Context question: (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}

? Context answer:

\vec{v}_{0} \| Y

\omega_{0}=0

Context question: (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. Context answer:

\vec{v}_{0}=-\frac{q d B}{2 m} \hat{j}

and

R_{c}=\frac{q B d}{2 m|\omega_{0}|}

and

(x_{c}, y_{c})=( \pm R_{c}, 0)

Context question: (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. Context answer:

2 \frac{q B}{m}