Spherical Shell in Space

[Arthur]

Please help me understand this problem. I am unable to solve it correctly.

An insulating uniform spherical shell of radius r and mass m is uniformly charged on its surface with a total charge of Q. Initially the spherical shell is at rest. Now, slowly apply a magnetic field in the direction of z in a large cylindrical space.

(1) Assume that the center of symmetry of the magnetic field is at the center of the sphere, so that the total external force on the spherical shell is 0. Find the angular velocity of the spherical shell when the magnetic field increases to B_0.
(2) Assume that the center of symmetry of the magnetic field is at a distance l_0 from the center of the sphere, so that the spherical shell will also be subjected to a resultant external force. Find the state of motion of the spherical shell when the magnetic field increases to B_0.

[Arthur]

Also, what does the state of motion mean in the 2nd part? Does it mean we need to find the angular velocity? I am assuming so.

[Alisher (Alisher Yerkebayev)]

@Arthur

Does it mean we need to find the angular velocity?

Yes, the sphere will rotate around its own axis and around the circle of the radius l_0 (it would be quite easy to prove that dl/dt = 0). In order to calculate these angular velocities, find the electric field distribution in space from \displaystyle\oint_l\vec E\cdot d\vec l = -\frac{\partial}{\partial t}\int_S\vec B\cdot d\vec S and integrate the force F acting on a sphere

In the same way, just calculate the net torque from the electric field. The homogenous magnetic field doesn’t affect the sphere’s rotation around its own axis but makes it undergo a circular motion (accelerated by E) around the magnetic field’s symmetry axis.

[Arthur]

Ok thanks, btw what answer are you getting for the 2nd part?

[Alisher (Alisher Yerkebayev)]

v=\frac{QB_0l_0}{m},\qquad\omega=\frac{QB_0}{2m}

[Arthur]

Thank you!