Time it takes for B to go to zero with R and L given
It is almost evil to answer your question as stated, but I'm going to do it anyway.

If you are talking about a coil with internal resistance that is shorted, or if you are talking about an explicit LR circuit, then it probably matches the schematic below.  The equation that controls the circuit is way simpler than you make out: $$ R i_L = -L\frac{di_L}{dt} $$

This quickly simplifies to \$\frac{d i_L}{dt} = -\frac{R}{L} i_L\$

This is a simple 1st-order linear ordinary differential equation with solution $$ i_L(t) = A e^{-t/\tau} $$ where \$\tau = \frac{L}{R}\$ and \$A\$ is found from some boundary condition on the differential equation.  For simplicity's sake, just say that we don't know what all else happened before \$t = 0\$, but at \$ t = 0\$ \$ i_L(0) = i_0\$.  Then $$ i_L(t) = i_0 e^{-t/\tau} $$

Now note that \$ e^x \$ is strictly positive for any finite value of \$x\$, and the answer to your question is -- never.  That's it!







simulate this circuit &ndash; Schematic created using CircuitLab