Problem $5$: A positively charged ion with charge $q$ and mass $m,$ initially at rest inside the source, is accelerated across a gap by an electric potential difference with magnitude $V.$ The ion enters a region of uniform magnetic field $B_1$ pointing into the page of the figure shown and follows a semi$-$circular trajectory of radius $R_1.$ The ion is again accelerated across the gap, increasing its speed, by an electric potential difference that has the opposite sign but has the same magnitude $|V|.$ The ion then enters a region of uniform magnetic field $B_2$ pointing into the page of the figure shown and follows a semi$-$circular trajectory of radius $R_2.$

a$.$ What is the ratio $\frac{B_2}{B_1}$ of the magnitudes of the magnetic field in the different regions in terms of the radii $R_1$ and $R_2$ and velocities in the two regions $v_1$ and $v_2?$

$\frac{B_2}{B_1}=$

b$.$ Considering the accelerating potential, what is the ratio $\frac{v_2}{v_1}?$ Express your answer in terms of $q,m,$ and $V$ as needed.

$\frac{v_2}{v_1}=$

c$.$ If $R_2=2R_1,$ what is $\frac{B_2}{B_1}?$ Express your answer in terms of $q,m,$ and $V$ as needed.

$\frac{B_2}{B_1}=$