Singly$-$ionized carbon atoms are accelerated through $960$ V and passed into a mass spectrometer to determine the isotopes present. The magnetic field strength in the spectrometer is $0\mathrm{.}240$ T $.$

$($a$)$ Determine the orbital radii for the ${?}^{12}C$ and ${?}^{13}C$ isotopes as they pass through the field.

$($b$)$ Show that the ratio of the radii may be written in the form

$\frac{r_1}{r_2}=\sqrt[\phantom{2}]{\frac{m_1}{m_2}}$

by verifying that your radii in part $($a$)$ satisfy this formula.

$\frac{r_1}{r_2}=$

$\frac{m_1}{m_2}$

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