The Problem states:
A metallic ring of Mass $M$ and radius $r$ falls freely under the influence of gravity in the direction along the negative Z-axis. A magnetic field $B_z = B_0(1-z\lambda)$ where $z$ is the Z coordinate of the center of the ring, also exists along the positive Z-axis. Initially , the center of the ring is at the origin. The resistance of the ring is $R$ .The plane of the ring is perpendicular to the Z-axis. Find the terminal velocity of the ring.
I figured, since the question is asking for a terminal velocity , there should be a force acting on the ring in the direction opposite to the gravitational force. But the force on a small segment of the ring , by the magnetic field due to the induced current would be radially inwards. Hence , the total force on the ring by the magnetic field would be zero.
My question is where would this force arise from?
Best Answer
The magnetic field through the ring changes as it falls, and whenever you have a conducting loop with a changing magnetic field through it, you get a magnetically induced current in the loop (remember Faraday's law). The current flows in the XY plane, perpendicular to the magnetic field, and a current flowing perpendicular to a magnetic field experiences a magnetic force ($\vec{F} = I\vec{\ell}\times\vec{B}$). That's going to be your upward force.
Now you at this point you "might be" (i.e. you are) wondering how $\vec{F} = I\vec{\ell}\times\vec{B}$ can ever produce a component of force in the Z direction, given that the magnetic field is in the Z direction and $\vec\ell\times\vec B\perp \vec B$. The reason is that $\vec{B}$ is not actually in the Z direction. It can't be, given the conditions of the problem. The magnetic field has to satisfy $\vec\nabla\cdot\vec{B} = 0$, but
$$\vec\nabla\cdot B_0(1 - z\lambda)\hat{z} = -\lambda B_0 \neq 0$$
The most plausible explanation is that whoever designed the question didn't think it through properly, but one could perhaps claim that the question didn't specify the X and Y components of the magnetic field, and thus there are nonzero components in (at least one of) those directions. These components will produce the vertical force on your ring.
If there are not nonzero components of $\vec{B}$ in the X and/or Y directions, then the magnetic field has to be uniform, and in that case there will be no upward force.