10.1.11. A particle of mass $m$ and charge $q$ enters a uniform magnetic field of induction $B$ at an angle $\alpha$ to the field with velocity $v$. Find the radius and pitch of the helical path along which the particle moves.
Solution
Let us decompose the particle's velocity into two components — parallel to the magnetic field: $v_{\parallel} = v\cos\alpha$ and perpendicular to the magnetic field: $v_{\perp} = v\sin\alpha$.
The perpendicular component causes circular motion due to the action of the Lorentz force, while the parallel one causes uniform translational motion along the magnetic field lines. The combined action of these two motions forms a helical path.
Let us write Newton's second law: $$qv_{\perp}B = m\frac{v_{\perp}^2}{R}$$ $$R = \frac{mv_{\perp}}{qB} = \frac{mv\sin\alpha}{qB}$$
The period of revolution (time of one turn): $$T = \frac{2\pi R}{v_{\perp}} = \frac{2\pi \frac{mv\sin\alpha}{qB}}{v\sin\alpha} = \frac{2\pi m}{qB}$$
The pitch of the helical path (distance along the field per one turn): $$h = v_{\parallel} \cdot T = v\cos\alpha \cdot \frac{2\pi m}{qB} = \frac{2\pi mv\cos\alpha}{qB}$$