Critical Helical Convergence

Imgur Let us consider a gaussian surface ABCD a cylinder shown as a rectangle.

The flux through the surface can be related with field by gauss law of magnetism or the maxwells equation. $\oint B\cdot dS=0$

$\intop^{CD}B\cdot dS=\intop^{AB}B\cdot dS+\intop^{ABCD}B\cdot dS$ ABCD represents the curved surface while AB and CD the base let $\frac{\mu_{o}nI}{2}=c$

$c\pi r^{2}=\int_{0}^{z}2\pi rdzB_{r}+c(1-\frac{z}{\sqrt{z^{2}+R^{2}}})\pi r^{2}$ Applying newton's leibnitz differenciation we get $B_{r}=\frac{c}{2}\frac{rR^{2}}{(R^{2}+z^{2})^{\frac{3}{2}}}$ As the magnetic force doesn't do any work the speed remains the same $V_{o}=\sqrt{V_{1}^{2}+V_{2}^{2}}=\sqrt{V_{1_{0}}^{2}+V_{2_{0}}^{2}}$ $-m\frac{V_{1}dV_{1}}{dx}=B_{r}V_{2}q$ Integrating and applying appropriate limits we get $V_{2}=V_{2_{o}}+\frac{qcr}{2m}\frac{x}{\sqrt{x^{2}+R^{2}}}$ If the particle has zero horizontal velocity on its path then it will return back in a helical path . for this $V_{0}=V_{2}$ $V_{0}=V_{2_{o}}+\frac{qcr}{2m}\frac{x}{\sqrt{x^{2}+R^{2}}}$ $V_{1}<\sqrt{(\frac{qr\mu_{o}nI}{4m})^{2}+V_{2_{0}}(\frac{qr\mu_{o}nI}{4m})}$

If the horizontal velocity exceeds this then the particle will go to infinity and not return.