Solenoids and Toroids — Question 9

Question 9

A solenoid current can be chosen from the required helical-orbit clearance

A long air-core solenoid has turn density n=800 per metre and clear bore radius a=.0200 m. A nonrelativistic positive ion has mass $m=4.00\times10^{-26}$ kg, charge $q=1.60\times10^{-19}$ C and transverse speed $v_\perp=2.00\times10^4$ m/s. Its helical guiding center is on the axis. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Find the minimum solenoid current for the orbit radius to be no greater than a.

2. At this limiting current, find the axial pitch for $v_\parallel=3.00\times10^4$ m/s.

Question 9 - Solution

A solenoid current can be chosen from the required helical-orbit clearance

(a) Method and calculation

$$r_L=\frac{mv_\perp}{qB}\le a,\quad B=\mu_0nI\Rightarrow I\ge\frac{mv_\perp}{q\mu_0na}.$$$$B_{\min}=.250\,\mathrm T,\quad\boxed{I_{\min}=248.680\,\mathrm A}.$$

(b) Interpretation and further calculation

$$T=\frac{2\pi m}{qB_{\min}}=2\pi\frac{a}{v_\perp}=6.28319\,\mu\mathrm s.$$$$p=v_\parallel T=\boxed{.188496\,\mathrm m}.$$

Physical check

The guiding-center condition matters: injection directly on the axis with transverse velocity would place the orbit center off axis and require a different clearance condition.