Question

1. An infinitely long solenoid of radius R carries n turns per unit length and current I and is oriented such that its axis is along the z direction. What direction must a particle of charge q be moving such that it feels zero force (a) inside the solenoid (b) outside the solenoid

2. Find the maximum magnetic flux through a circular coil of radius L such that L < R placed inside the solenoid of the previous problem.

Answer 1

The magnetic field due to an ideal solenoid of radius R and carrying current I is
   
And force acting on the particle due to a magnetic field is
  
a)
So, given the charged particle, the force will be zero if the cross product is zero and so, the vectors should be parallel. So, the particle should move parallel to the z axis, i.e,   inside the solenoid.
b)
And as there is no magnetic field outside the solenoid, so, the particle can move any direction and the particle will not feel any force.

2)
The flux through the circular coil of radius is
   

where, \theta is the angle between the magnetic field and the normal to the surface of the coil. And A is the cross sectional area of the circular coil of radius L.
So, the maximum magnetic flux is