C

To find the time after which the bead starts slipping, we first need to analyze the forces acting on the bead.
Step 1: Calculate the tangential acceleration
The rod is rotating with an angular acceleration \(\alpha\). The tangential acceleration \(a_t\) of the bead at a distance \(L\) from point A is given by:
\[ a_t = \alpha L \]
Step 2: Calculate the frictional force.
The maximum static frictional force (before slipping) can be expressed as:
\[ F_f = \mu m g \]
Since we are neglecting gravity, the frictional force will be assumed as the force required to hold the bead against the tangential acceleration.
Step 3: Set up the relation for slipping.
For the bead to start slipping, the tangential force must equal the maximum frictional force:
\[ m a_t = \mu m g \]
Since gravity is neglected, it implies that we need to check when the sliding begins based solely on the tangential acceleration.
Now, from equations: \( \alpha L = \mu \), we find time \(t\) in terms of angular acceleration and distance. Since the angular distance is related to angular acceleration as \(\theta = \frac{1}{2} \alpha t^2\).
Thus, upon solving, we find:
\[ t = \sqrt{\frac{L}{\alpha \mu}} \]
Therefore, the answer matches to option C.