Singly$-$ionized carbon atoms are accelerated through $1050$ V and passed into a mass spectrometer to determine the isotopes present. The magnetic field strength in the spectrometer is $0\mathrm{.}230$ T$.$

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

r$1=$ cm $(12$C$)$r$2=$ cm $(13$C$)$

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

r$1$r$2$

$=$

m$1$m$2$

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

r$1$r$2$

$=$

m$1$m$2$

$=$