Question 2.31: Calculate the Larmor radius for an electron and an argon ion...

Before calculating the Larmor radius for either particle, the velocity of each particle must be computed. Since the particles have gained an energy of    q _{ e } \Delta V =1 eV =1.602 \times 10^{-19} J , this energy will appear as kinetic energy and we wherite

q _{ e } \Delta V =\frac{ m _{ e } v _{ e }{ }^{2}}{2}=\frac{ M _{ Ar } v _{ Ar }{ }^{2}}{2}

  v_{e}=\sqrt{\frac{2 q_{e} \Delta V}{m_{e}}} \text { and } v_{A r}=\sqrt{\frac{m_{e}}{M_{A r}}} \sqrt{\frac{2 q_{e} \Delta V}{m_{e}}}

The atomic number of argon is 40 yielding a ratio of the masses to be

\frac{ m _{ e }}{ M _{ Ar }}=\left(\frac{1}{1836}\right) \times\left(\frac{1}{40}\right)

The Larmor radius for the electron is found from (2.142) to be

r_{j}=\frac{M_{j} v_{j}}{q B}      ( 2.142)

r_{e}=\frac{m_{e} v_{e}}{q_{e} B }=\frac{m_{e} \sqrt{\frac{2 q _{e} \Delta V }{m_{e}}}}{ q _{ e } B }=\frac{\left(9.1 \times 10^{-31}\right) \sqrt{\frac{2\left(1.602 \times 10^{-19}\right)(1)}{\left(9.1 \times 10^{-31}\right)}}}{\left(1.602 \times 10^{-19}\right)\left(10^{-2}\right)}=3.4 \times 10^{-4}   m

The velocity of the argon ion can be expressed in terms of the electron velocity

  v_{ Ar }=\sqrt{\frac{ m _{ e }}{ M _{ Ar }}} \sqrt{\frac{2 q _{ e } \Delta V }{ m _{ e }}}=\sqrt{\frac{ m _{ e }}{ M _{ Ar }}} v _{ e }

Therefore, the Larmor radius for the argon ion is found from (2.142) which can also be expressed in terms of the electron Larmor radius

\begin{aligned} r_{A r} &=\frac{M_{A r} v_{A r}}{q_{e} B}=\frac{M_{A r} \sqrt{\frac{2 q_{e} \Delta V}{M_{A r}}}}{q_{e} B}=\left(\frac{M_{A r}}{m_{e}}\right)\left(\frac{m_{e} \sqrt{\frac{2  q_{e} \Delta V}{m_{e}}}}{q_{e} B}\right)\left(\sqrt{\frac{m_{e}}{M_{A r}}}\right) \\ &=\sqrt{\frac{M_{A r}}{m_{e}}} r_{e}=\sqrt{40 \times 1836} r_{e}=9.2 \times 10^{-2}   m \end{aligned}

A comparison on the two Larmor radii indicates that the electrons are closely “tied” to the magnetic field lines and the ions are not. In several applications, the electrons are said to be “magnetized” and the ions are “unmagnetized.”