Can someone please clear this doubt regarding Lenz's Law?

Question

In Lenz's Law or Faraday's Law of Electromagnetic Induction, whenever there is a change in magnetic flux in a solenoid due to an external magnetic field like an approaching magnet there is an EMF generated, this EMF then generates a current and that current's magnetic field opposes the original cause of change of magnetic flux (which in our case is an approaching magnet).

This can be formally stated as follows: $$\textrm{emf}_b = -N\frac{d\phi}{dt}$$ I understand the basic concept but I have a few doubts. Let's say the magnet approaches the solenoid with instantaneous velocity $v_0$ at some time $t_0$. Since the change in flux is happening we will apply the above equation to calculate the EMF value ($\textrm{emf}_b$). Then we can calculate the current by ohm's law: $I=\textrm{emf}_b/R$. This $I$ provides the necessary opposing magnetic field to push back the approaching magnet.

Doubt 1: If there are two objects pushing at each other with equal amount of force there should be no motion at all? But be know that the approaching magnet slows down or stops, but the motion happens. Why is that? Is it due to the fact that there are power losses in the solenoid too given by $P_l= I^2R$. Is that what is creating the imbalance?

Doubt 2: If we put in an iron core in the solenoid there would be eddy currents (if I am correct) and these eddy currents too would cause some losses. Does that mean the capability of the solenoid to 'push back' the approaching magnet decreases even further, creating more imbalance?

Doubt 3: If I reduce $R$ exponentially (say by increasing the cross section area of the wire) can I make the current's arbitrarily large?! The equation says so. but that means I can increase the power losses as much as I want with same/limited EMF?! I am a bit unclear but that shouldn't be possible.

Can someone please clarify where my understanding is wrong (I am sure it is somewhere).

Answer 1

Doubt 1: E&M is a conservative force. In this context that means linear momentum is conserved. So the force exerted on the approaching magnet is exactly matched by an opposing force on the coil. There is no imbalance. Example: When an electric motor turns its rotor with torque T, the body of the motor experiences an exactly balancing torque -T. The balancing torque has to be resisted by the structure holding the motor.

Doubt 2: Iron cores for a solenoid will be a complicated situation. The cores usually mean the solenoid is stronger, so an approaching magnet probably feels stronger force. But the flux will be distorted by the core, so the exact result is complicated and depends on the details. However, there is still no imbalance in force. The solenoid feels exactly the reverse of the force the magnet feels.

Doubt 3: Larger radius wire does have lower resistance, but it's not possible to go to zero. The resistance is (at least to first approx) proportional to the inverse of the area. So by increasing the radius by a factor of 10 you get 1/100'th the resistance. However, a wire 10 times as thick is harder to wind on a coil. You get less turns per meter. So you can't get arbitrarily high interaction with a magnet. You can get very much higher than, say, a thin copper wire. There are a number of demos of this. Here is one on YouTube. And another.