Charged pendulum in magnetic field

[Muthumanimaran]

Homework Statement

The question is in the attached document

Homework Equations

Newton's second law states that F=ma
Charged particle in magnetic field experiences F=Q ( v X B)

The Attempt at a Solution

Since the charge 'Q' is constrained to move along a path in xy-plane in such a way that the distance between origin and the bob is "l", the velocity of the Q can be resolved in horizontal and vertical components. Since the horizontal component of velocity is in the direction perpendicular to the direction of magnetic field it also experiences Lorentz force along the direction to xy-plane.

$$F=-mg\sin(\theta)+BQv$$
Am I going in the right way? Do I need to find the solution for the above differential equation to know the equation of motion. If I take the derivative of equation of motion with theta and equate it to zero, will it give the minimum value of theta in this problem? Am I thinking in the right way? Or is there an elegant alternate approach to this problem?

 

[berkeman]

Since the charge 'Q' is constrained to move along a path in xy-plane

The problem statement says the B-field points "up" in the positive z direction, but your statement seems to imply that you are taking "up" as the positive y direction. Can you clarify what the coordinate axes are for this problem? Is there a figure or diagram that goes with the problem? Thanks.

 

[drvrm]

Am I going in the right way? Do I need to find the solution for the above differential equation to know the equation of motion. If I take the derivative of equation of motion with theta and equate it to zero, will it give the minimum value of theta in this problem? Am I thinking in the right way? Or is there an elegant alternate approach to this problem?

i was just thinking aloud ...if the magnetic field is acting on the charged bob it will give a force which is perpendicular to velocity i.e. it can not do any work..so the energy of the bob should be as its in gravitational field..so how you get a minimum angular displacement.

Moreover i agree with @ Berkemn's comment that the direction of the magnetic field may be clarified...the term vertically up..

 

[berkeman]

if the magnetic field is acting on the charged bob it will give a force which is perpendicular to velocity i.e. it can not do any work..so the energy of the bob should be as its in gravitational field..so how you get a minimum angular displacement.

Agreed. Maybe the pendulum rod is free to move in more than just a plane (ball hinged top?)...