A particle with mass \(m\) and positive charge \(q\) enters a uniform magnetic field \(\vec{B}\) with velocity \(\vec{v}\). The velocity vector makes an angle \(\theta\) (where \(0 < \theta < \dfrac{\pi}{2}\)) relative to the direction of \(\vec{B}\), causing the particle to execute a helical trajectory. Which of the following expressions correctly represents the radius \(R\) of the particle's helical path, and by what factor does the pitch \(p\) (the distance traveled parallel to \(\vec{B}\) in one full revolution) change if the magnetic field magnitude is doubled?