Solution

In addition to gravity, the conductor experiences upward magnetic force $F=IlB$, where $I$ is the current through the conductor. The current flows in the circuit, because there is an induced EMF $\mathcal{E}=Blv$, where $v$ is the downward speed of the conductor, as a result of increasing area of the circuit loop and hence magnetic flux.

In the case when the circuit is closed with the resistor, $I=\mathcal{E}/R=Blv/R$, and the equation of motion of the conductor becomes

$$m\dot v=mg-B^2l^2v/R.$$

Subject to the initial condition $v=0$ at $t=0$, we have the solution

$$v=\frac{mgR}{B^2l^2}(1-e^{-B^2l^2t/(mR)}).$$

In the case when the circuit is closed with the capacitor, $I=C(d\mathcal{E}/dt)=CBl\dot v$, and the equation of motion of the conductor becomes

$$m\dot v=mg-CB^2l^2\dot v\qquad\Rightarrow\qquad(m+CB^2l^2)\dot v=mg.$$

Subject to the initial condition $v=0$ at $t=0$, we have the solution

$$v=\frac{mgt}{m+CB^2l^2}.$$