A cylindrical conductor of outer radius \(R\) carries a pulse of current directed parallel to its central axis.

The conduction electron density varies with distance \(r\) from the central axis according to \(n(r) = n_0 \left(\dfrac{r}{R}\right)\), and all electrons move with a time-dependent drift speed \(v_d(t) = v_0 \left(1 - \dfrac{t^2}{T^2}\right)\) for \(0 \le t \le T\), where \(n_0\), \(v_0\), and \(T\) are positive constants and \(e\) is the elementary charge.

Which of the following expressions correctly represents the total charge \(Q\) that passes through the cross section of the conductor in the region \(\dfrac{R}{2} \le r \le R\) during the time interval from \(t = 0\) to \(t = T\)?