Coils of wire with current in a magnetic field

[Oijl]

Homework Statement

The figure shows a wooden cylinder with mass m = 0.100 kg and length L = 0.800 m, with N = 20.0 turns of wire wrapped around it longitudinally, so that the plane of the wire coil contains the long central axis of the cylinder. The cylinder is released on a plane inclined at an angle θ to the horizontal, with the plane of the coil parallel to the incline plane. If there is a vertical uniform magnetic field of magnitude 0.250 T, what is the least current i through the coil that keeps the cylinder from rolling down the plane?

Homework Equations

The Attempt at a Solution

I know I want a total net force of zero acting on my cylinder. This is a list of my thoughts on the problem so far:

I believe (I know) that, ignoring the wires and looking just at the wood, there is a net force on the cylinder such that it will move down the ramp.

I believe (I am very sure) that when I sent a current i through the wires, a force is produced acting "right" on the wires with the current moving away from the viewer, and a force is produced acting "left" on the wires with the current moving towards the viewer.

I believe that sending a current i through the wires will produce a net force of zero acting on the wires.

I believe that therefore the net force on the cylinder/wire object will not change when the forces due to the magnetic field are considered.

I believe (but know is false) that therefore the cylinder will move down the ramp.

I believe (I pretty much know) that the forces due to the current through the field produce a torque on the loop of wires around the cylinder - that this torque would rotate the loop until the normal vector of the loop pointed in the same direction as the magnetic field.

Since no coefficient of friction is given in this problem, friction is not to be considered.

I believe that therefore the cylinder can rotate due to torque without producing a new force that would resist its movement down the ramp.



So how can producing a net force of zero on an object that doesn't already have a net force of zero result in a net force of zero?

 

[rl.bhat]

whether θ is given in the problem?

 

[Oijl]

Theta is not given. I assume that in the correct process of finding the current required to create equilibrium, theta is related to the component forces produced by the current-carrying wires. Since theta is also related to the component forces due to gravity, I figure theta will cancel out.

 

[Oijl]

Nevermind, I solved it.