A flat circular coil consists of \(N\) tightly wound turns of wire, has radius \(r\), and has total electrical resistance \(R\). The coil lies initially in the \(xy\)-plane within a uniform, constant magnetic field \(\vec{B} = B_0 \hat{k}\). The coil is then rotated by \(180^\circ\) at a constant angular speed \(\omega\) about a fixed diameter parallel to the \(x\)-axis. Which of the following expressions represents the total magnitude of electric charge \(Q\) that passes through any cross-section of the wire during this rotation?