Question #43006

A proton of mass $m$ and charge $q$ enters a region of non-uniform static magnetic field. At the origin ($z=0$), the magnetic field is $\vec{B} = B_0 \hat{k}$ and the proton has a velocity $\vec{v} = v_0 \cos(30^\circ) \hat{k} + v_0 \sin(30^\circ) \hat{i}$. The magnetic field points primarily in the $+z$ direction, and its magnitude slowly increases along the $z$-axis. Assume the spatial variation of the magnetic field is gradual enough that the magnetic moment $\mu = \frac{m v_{\perp}^2}{2B}$ remains an adiabatic invariant. Find the ratio of the local pitch of the proton's helical path at $z=0$ to its local pitch at a point where the magnetic field magnitude becomes $2B_0$. (The local pitch is defined as $p = v_{\parallel} T$, where $v_{\parallel}$ is the longitudinal velocity and $T$ is the local time period of gyration).