Finding the induced current in a loop and force acting on the conductor 
The conductor has a velocity to the right and is part of a closed loop (see the picture). Find the direction of the induced current and the direction of the magnetic force on the conductor
There must be induced a magnetic field going into the plane of the paper to counteract the increase in flux going out of the plane of the paper. The force must be going in the opposite direction of the velocity, so using the right-hand rule: straight fingers pointing upwards through the conductor, curled fingers down and thumb to the left, giving a current going counterclockwise. Why is this not correct?
When it comes to the force, we know it must be going in the opposite direction of the conductor (Lenz' law), but what if we wanna find it using the right-hand rule? To get that right i have to use that the current goes clockwise (which is correct), but now i have to use the exterior magnetic field to get the force right? Why is this? Why do i have to use the induced magnetic field when finding the induced current, but when im finding the induced magnetic force, i have to use the exterior magnetic field. Why?
In addition, could i use that the direction of the charges is to the right, and use that to find the direction of the current? Whats the difference between a force acting on the conductor, and a force acting on electrons inside the conductor?
 A: Regarding the external versus induced field question, one could argue that the induced field wraps around the conductor and any forces cancel, but other more distant parts of the loop could produce forces on the conductor. This issue makes teaching the field concept difficult. Regarding the distinction between the force on the "conductor" and the force on the electrons, there was an excellent article in the American Journal of Physics a few years back explaining how the "ILB" force is actually electrostatic resulting from a Hall effect electric field resulting from the magnetic deflection of the electrons. The ions, which make up almost all the mass of the conductor, interact with this field via the F=qE component of the Lorentz force.
A: We start with the "exterior" field. When charges inside the conductor move relative to this field, they feel a force and are deflected. This causes a current to flow.
Now there is a current, there is a component of velocity of the charges that is perpendicular to the motion of the loop. And this is the origin of the force.
While it is convenient to say "Lenz's Law helps me figure out the direction of the induced field so I know what direction the current is flowing", there is no problem with saying "the charge is moving relative to the external field, and this is the resulting force and therefore the resulting direction".
The two approaches will give the same result.
A: An external force is required to keep the rod moving to the right because the interaction of the induced current with the external magnetic field produces a drag force which acts to the left. The induced magnetic field inside the loop will be opposite to the external field because it is trying to offset the increasing flux through the loop (as the loop gets larger).  The effect of the induced field on the rod will be complex because the rod is acted on by field contributions from the other three straight segments of current carrying wire.
