The 2 Forces on the Conductor having induced Motional EMF

Question

Ok, a conductor PQ is placed on a U-shaped conducting rails situated in a magnetic field, B directed inside the plane of paper. When it is moved to left with velocity ,v, the free electrons will experience a Lorentz Magnetic Force,

F = qvB where q,v=Charge and velocity of the free elctrons

This force, F will drive free elctrons to point P(as per Fleming's Left Hand Rule) and accumulate same positive charges on point Q

Now the force on the movable arm PQ is given by

F' = ILBsin90 where, I=Induced Current L= Length of Conductor

This force, F' will act to the opposite direction of velocity, v.Hence same amount of external work is needed to be done to move the arm to left(as ler Fleming's Left Hand Rule).

Now, the question is that in general, the force ILBsin(theta) is derivable from the force, qvB and I think both are magnetic Lorentz force. Also, generally, both the forces have same direction.

But here in this case both the forces have different direction. Do both are different.If yes, how?

If both the forces here are same then, how could that be possible?

Answer 1

Because a conductor $PQ$ containing mobile charges $q$ is moving with a speed $v$ at right angles to an magnetic field $B$ the mobile charges have a force exerted on them along the conductor in the direction from $P$ to $Q$ if the mobile carriers are positive and in the opposite direction if the mobile carriers are negative. The charges in that conductor move under the influence of that force and since there is a complete circuit an induced (conventional) current $I$ flows in the circuit and through the conductor from $P$ to $Q$.

The conductor $PQ$ having a current passing $I$ passing through it in a magnetic field $B$ at right angles to the current has a force on it in the opposite direction to that in which it is moving, ie in the oppose direction to $\vec v$, to the right.
So to keep that conductor moving an external force to the left must be applied on the conductor.
This is Lenz's law in action.
The direction of the induced current is such as to oppose the motion producing it which in this case is conductor $PQ$ moving to the left.

If the conductor had a force on it to the left it would either speed up the conductor, increase the induced current, heat the wire, do some electrical work, etc or allow you to be dragged along with the conductor with the conductor doing work thus contravening the law of conservation of energy.