Problem $1-$ Magnetic Fields and Forces $(20$ pts$)$

A negatively charged particle with charge, $\backslash (-$q $\backslash ),$ and mass, $\backslash ($ m $\backslash ),$ enters a region $($shown bounded by the dashed and solid line$)$ with a magnetic field, $\backslash (\backslash$mathrm$\{$B$\}\_\{0\}\backslash ).$ You want to get the particle through a hole that is a distance a from where it enters the field. All answers should be with respect to $\backslash ($ q$,$ v$\_\{0\},$ B$\_\{0\},$ m $\backslash ),$ and $\backslash ($ a $\backslash ).$

a$)$ What is the direction of the magnetic field in this region? $(3$ pts$)$ How high above the surface that has the hole does the particle enter the field $($i$.$e$.,$ what is b$)?(2$ pts$)$

b$)$ What is the force $($magnitude and direction$)$ acting on the particle as it enters the field? $(5$ pts$)$

Force $($magnitude and direction$)$

c$)$ What is the velocity, $\backslash ($ v$\_\{0\}\backslash )$ of the particle in terms of $\backslash (\backslash$mathrm$\{$q$\},\backslash$mathrm$\{$B$\}\_\{0\},\backslash$mathrm$\{$~m$\}\backslash ),$ and a$?(5$ pts $)$

d$)$ What would the velocity need to be to get the particle through the hole if the mass were half that in the previous parts $($i$.$e$.,$ mass $\backslash (=\backslash$mathrm$\{$m$\}/2\backslash ))?(5$ pts$)$